My son and I usually drive to Tampa once a year to watch some baseball for a week. We head there when the Rays play the Yankees (our favorite team) or when the Angels are in town so he can see Trout play. We also catch some Tampa Tarpons games at Steinbrenner field in hopes of seeing a future Yankee.
Personally, I don't consider Tropicana Field a dump. It's just older and doesn't have all of the bells and whistles of the new ballparks. All of the staff I have encountered at the Trop are above and beyond fantastic. And for the kids, the players are very good at signing autographs before every game. Unfortunately, NONE of the visiting teams we have seen will sign autographs before games. This even happens on the Thursday afternoon games when there is no pre-game BP on the field. We site first or second row visitor dugout (only about $90 per ticket) and Trout, Judge, and all the others won't even make eye contact with the kids.
Even when the Yankees are in town they don't sell out, although attendance is higher those games. I'd also say 2/3 of the people in the stands are Yankee fans.
For the Tarpons (formerly Tampa Yankees) games, Steinbrenner Field is a beautiful place to watch a game. But the stands are nearly empty every night. I don't think I've ever seen more than 1,000 fans there. But we love to go there and most of the players are really great with the kids.
So for whatever reason, the fans don't seem to show up at any of those games. But I'm OK with that because it means we get cheap ticket prices whenever we head down there.