In my time, it was transistor am radios, but yes, they brought them.
The transistor radios were something of a Dodger tradition from when the team first landed in Los Angeles
but it started out of perverse of necessity: While awaiting the completion of Dodger Stadium, the Dodgers played in the
old Los Angeles Coliseum. It had to be seen to be believed, how they shoehorned a baseball field into that big football oval . . .
. . . and for fans who had seats far behind the outfield, it wasn't easy to see the game action. So
they brought in their portable radios to listen to the broadcasts. And Vin Scully became as big a
star as the Dodgers' players might become as a result. So much so that when Dodger Stadium
opened fans kept bringing their radios (and, in due course, picture-blacked portable TVs) anyway.
As for the Coliseum, in due course it played a role in the O'Malley family selling the Dodgers in the 1990s:
Peter O'Malley wanted to build an NFL stadium on land the O'Malleys already owned; O'Malley believed
(and he probably wouldn't have been wrong) that the revenues from that stadium (I could be wrong,
but I think he would have leased it to an NFL team and split the parking and concessions with them)
would have enabled him to keep and keep running the Dodgers. Los Angeles's powers that be absolutely
insisted that the only way the NFL would come back to Los Angeles would be if they agreed to play in the
antique Coliseum. Peter O'Malley threw up his hands and was forced to sell the Dodgers at last because
he didn't want his family to be hit with onerous estate and inheritance taxes if he simply transferred
the team to one of his siblings or children.
Further sad irony: When Frank McCourt was being forced to sell the Dodgers after it turned out he used
the team as his personal ATM machine while cutting drastically on necessary expenses like stadium
security and the like, it turned out that Peter O'Malley was one of the people trying to put together
a group to buy the team. (For years, fans who met O'Malley asked if not begged him to find a way
back to the Dodgers, while first News Corp. and then McCourt owned the team.)
Today, O'Malley is sort of back in baseball: his family co-owns the San Diego Padres. His two sons
and two nephews own the team and he is said to act as a consultant.