Don't be an idiot. It is far more expensive to militarily "seal" a 2,000-mile long border than to hire additional folks to vet asylum seekers and monitor the locations and activities so they show up for their hearings. Trump wants $25 billion for his wall. How about YOU pay for it?
I don't think that's true. "Sealing" the border is basically a red herring. A $25B wall, backed by a reasonable number of border patrol, can reduce the flow of illegal immigrants massively.
In contrast, unless you support holding all those asylum seekers in detention where they can be prevented from moving about inside the country, trying to track everyone who came in would be far more expensive, and would be a never-ending problem.
Then, when they are released into the U.S. pending the formal asylum hearing, I'd think they ought to be treated like folks on parole, obliged to check in with government officials on a weekly or monthly basis, so their locations and activities can be monitored. Violation of such requirements should be, without more, grounds for deportation and rejection of their asylum bid. Asylum seekers should be required to be fingerprinted and to sign agreements so as to allow expedited deportations in the event of violation of the "parole" requirement.
This is basically what happens already. And 90% of them simply "skip parole" and disappear into the country. They then use fake names, and that's it. Maybe some fraction of them are eventually caught and given the boot, but the overwhelming majority aren't. As a practical matter, it is either detention, or let them come in freely. There is no effective middle ground possible.
The only way to perhaps prevent that is by introducing and implementing a lot more biometric scanning for employment, etc., though it would necessarily have to apply to citizens as well. And even if you could pass that legislation tomorrow, it would takes years to implement all the technology and infrastructure to support it, which makes it an irrelevancy in terms of how we handle these people currently.