I do a bit of this for the local minor league team here. There is really a lot of things they can do to improve attendance.
Lower ticket prices in April and May and try to get some of that Disney World traffic to flow over to the ball park. The fans will be pumped to see a full season of Baldelli, Delmon, Upton and gang so attendance should be good to start. Lower the prices, fill the stands, and then hope we are a competitive, exciting team.
Get a huge advertisement program going through some of the local newspapers to get more attendance by using the discounts in April and May along with group packages for the families that will want to go. Maybe even have a thing with the local high schools where A/B students get 2 free tickets.
Have a huge shirt give away day. This will be a bit expensive, but imagine how the stands would look with the green on all those fans. The players would be pumped along with the DRO.
Have a meet and greet for the kids for only a few bucks with the more popular players once a week. Delmon, Baldelli, and Kazmir would have to be a must at least once.
Florida is a vacation, relaxation state, so they need to turn the ballpark into a relaxing environment. It is indoors, so use the "no bad weather" to a plus.