baseball cards are now owned by the Library of Congress and most have no copyright restrictions on record. This, however, does not mean that there are no trademark restrictions. Copyrights expire, but trademark rights don’t. Are there rules, law cases that give guidance on the use of such images where copyright and trademark rights may conflict?
Let's start with the principle that you can reproduce the card imagery for any informational purpose you want -- for example, in a book or a documentary or at a website explaining the history of baseball.
What if you sold a t-shirt? What if you sold merchandise with PD baseball card imagery that says "Reds" outside the Great American Ball Park? Would that trigger a lawsuit? Possibly. Would you prevail? We'd like to think you would. It's going to come down to a variety of factors such as the prominence of the name on the merchandise, consumer motivation for purchasing (Reds fan vs. baseball history fan), and issues similar to those raised in Dastar Corp. v. Twentieth Century Fox Film Corp (in which the Supreme Court held that a trademark claim could not be used to bar a public domain reproduction). Note, that case is not exactly on point with your situation and to see why, check out this article (scroll down to "C. Effect of Dastar on Merchandising Cases and on Federal Trademark Law.")
When the cards don't include the name of the team. Many of the cards in the collection don't include the names of the teams. As a general rule, if you're not using the name of the team, just the city, there shouldn't be any trademark issues (unless the lettering for the city/team is so distinctive that it makes an association with consumers.)
Teams that don't exist. Obviously for those baseball teams that are no longer in existence -- for example, the Boston Beaneaters -- the trademark rights are no longer being exploited and the marks are considered to be abandoned (although you may want to check the USPTO records to make sure nobody has resurrected these as zombie marks). As for the Brooklyn Dodgers, that mark had been abandoned and then brought back for certain merchandise (according to the ruling in Major League Baseball Properties, Inc. v. Sed Non Olet Denarius, Ltd). Nowadays, it appears that the mark has been resurrected and officially licensed for clothing (although non-infringing homages are always possible, too).
Bottom Line. If you don't want to get hassled, avoid merchandise that prominently displays existing team marks.