Big business types tend to support raising the minimum wage. Few of them actually pay a minimum wage. That's usually the realm of the small businesses. Raising the minimum wage is one way for big business to eliminate competition by raising overhead. The gained profits from cutting off (and redirecting) the competition make up for the increased overhead.
There is a counter-argument to be made in favor of the minimum wage, at least in principle: the additional transaction costs associated with a minimum wage are less than the transaction costs associated with the lack of such a wage, particularly those flowing from unionization. That, by itself, doesn't tell us how high or low a minimum wage should be, but it does mean that a legally-mandated minimum wage is a good thing to have.
The bottom line is that raising the minimum wage is only worth the while if the additional costs it imposes are still less than those imposed by what would happen in the absence of any minimum wage. That determination cannot be made using general principles but is wholly fact-dependent. Thus, whether it's a good idea to raise the minimum wage to $10.10 requires justification based on the current facts and circumstances and an analysis of the effects of that higher minimum wage relative to the costs of having no minimum wage at all. And for that we cannot evaluate Romney's statement unless we know his underlying justification for that statement.
Every employment transaction - the exchange of labor for consideration in the marketplace - requires agreement on the compensation to be paid, and that in turn requires negotiation. For most employment situations that process in and of itself doesn't prevent any significant number of such transactions. This follows, in part, because the information asymmetry between the seller of labor and the buyer of labor is not particularly significant; it exists, but has more to do with the would-be employer's actual labor needs (e.g., whether the position will end up entailing work that has little to do with the position as advertised), and less to do with the value of the labor (i.e., the compensation the hypothetical employer, under no compulsion to hire, is willing to pay), which is generally well-known and, in any event, because the would-be employee has skills that are in relative scarce supply, the employee's threat to walk away has teeth (relative, for example, to the skill-set possessed by unskilled people, which are largely the same skills everyone possesses no matter how highly educated or skilled otherwise).
However, for employment situations at the low(er) end of the economic spectrum, e.g., the employment of relatively unskilled, inexperienced individuals in so-called entry-level positions, the transactional costs involved may be high enough to prevent a significant number of otherwise good transactions. Furthermore, the relative bargaining power of the would-be employee is so minimal compared to that of the would-be employee that the employee's participation may be only quasi-voluntary.
A typical response to that asymmetry is unionization. However, as has been amply demonstrated by reality, unionization, particularly when coupled with government intervention, ends up creating substantial inefficiencies, and engenders substantial rent-seeking behaviour by unions and union members. Unionization also creates greater unemployment on its own because (a) the benefits of unionization aren't enjoyed equally by all similarly-situated individuals - those who have not, or cannot, unionize are put at a disadvantage and in fact may be barred from employment altogether where closed shop unions operate; and (b) the rent-seeking behaviour unions engage in results in an inefficient allocation of capital as union members are able to force employers to pay more than the actual market value of the members' labor - which means that there is less capital left over to hire non-unionized individuals at prevailing market rates. Finally, unionization, particularly where the government intervenes to favor unions over employers, creates substantial bargaining asymmetry - this time in favor of the employees over the employer, and can make the employer's participation in the market for labor only quasi-voluntary: the employer may be - nay, often is - forced to employ a significant number of individuals it would never voluntarily employ.
Setting a minimum wage obviates the need to negotiate a wage for each employee and puts a limit to the bargaining asymmetry that favors the employer. A minimum wage also reduces the market incentive to unionize, thereby reducing the economic costs of rent-seeking activity by unions. It also results in a more equitable distribution of the benefit amongst all similarly-situated individuals because it doesn't depend on the accident of membership, or not, in some particular organization. By doing so, a minimum wage reduces one of the bigger transaction costs that hinders the efficient market exchange of labor for compensation.
Of course, since a legal minimum wage is also an instance of government interference in/distortion of the market, it brings its own set of transaction costs. That cost is basically the number of foregone employments where an individual cannot find employment because the market value of that individual's skills is less than the minimum wage.
On balance, however, the costs created by a government-imposed minimum wage should be less than those created by unionization because a minimum wage creates much less rent-seeking behaviour - for example, an employer cannot be forced to employ individuals it would never voluntarily employ - and because a minimum wage results in less additional unemployment than does unionization because it (a) causes less misallocation of capital than does unionization (e.g., by reducing the amount of rent-seeking by employees) and (b) applies to all employees equally, reducing the unequal advantage enjoyed by some employees over other employees.
Given this, the real issue raised by a legally-mandated minimum wage is not whether it exists at all, but the appropriate level to set it at. If the minimum wage is significantly higher than the overall market value of the labor in question, then it starts to impose transaction costs approaching the same imposed by unionization. If it's set significantly lower than the overall market value of that labor, then it reinforces the same transaction costs that engendered unionization because it fails to significantly reduce the need for each individual to negotiate wages, and increases the bargaining asymmetry in favor of the employer because there is little incentive for an employer to bid higher than that minimum wage for the labor of its employees.
There are plenty of other areas of economic life where a similar cost-benefit situation exists - where the costs associated with the government's interference in the market are outweighed by the costs caused by the problems of price discovery and other informational transaction costs and the problems of bargaining power asymmetry.
For example, the original FDA was a reasonable response to the transaction costs associated with a "natural" market in food and drugs. How so? Because without those original FDA standards and their enforcement, every buyer of food or drugs would have to either (a) carry a chemical-testing kit with them to determine in each instance whether the food/drug being offered for sale contains adulterants that buyer does not wish to pay for, or (b) take the risk that any such adulterants will make him/her sick, requiring him/her to either incur additional costs by having to pay for medical care or pay for insurance against any such costs. The original FDA, by requiring disclosure of the ingredients of food/drugs offered for sale and prohibiting the addition of certain types of adulterants - those which can cause serious illness but are hard to discover at the time of purchase - certainly increased the cost of food and drugs, thereby preventing some otherwise economically efficient transactions, but the loss of those transactions was outweighed by the transaction costs arising in an unregulated market for food/drugs.