You should consult your homeowner's insurance policy or an experienced local property law attorney concerning your remedies. You should also check if the CC&Rs of your community will permit you to install netting around your back yard, as this might be your best protection.
Florida law provides that "living on a golf course and living with golf balls necessarily go hand-in-hand. The issue here is whether [you] are being subjected to more than a reasonable exposure to golfballs and what steps, if any, would be appropriate to remedy this problem." Bechhold v. Mariner Properties, Inc. 576 So.2d 921 (Fla. 2 DCA 1991). If you are being subjected to an unreasonable amount of golf balls then it is incumbant on the golf course to remedy the problem. If they fail to take appropriate steps then they may be liable to you for your damages on a theory of private nuisance. I would put them on written notice that they must immediately correct the problem. If they don't comply I would retain counsel to discuss filing a private nuisance action against them.
Attorneys Rose and Willis both gave good answers. Mr. Rose focused on practical "home improvements" that may be made to protect the home, while Mr. Willis focused on FL case law and potential causes of action.
The Mariner case cited by Mr. Willis provides an excellent discussion of the issues. Another worthwhile case is Jesters v. Taylor, 105 So.2d 569 (Fla. 1958). The Jesters case states that in determining whether the country club was negligent (as to a caddy being struck by a golf ball) the question is whether it had fulfilled its duty to exercise reasonable care to maintain the premises in a reasonably safe condition.
In the Mariner case, the golf course owner used various design features to minimize errant golf balls, including bunkers/mounds, a sign advising golfers to use "extreme caution" to avoid hitting balls into a residential area, as well as planting trees and shrubbery. Some or all of these steps could potentially be taken by your golf course owner to further protect your home.
Additionally, the plaintiff homeowner in the Mariner case had made home improvements to protect his home, including installing "bulletproof glass" where he'd experienced broken windows and "heavy-duty wire mesh" to protect his screen enclosure. Those are steps you too could potentially take, although they might require Association approval or building permits.
The current golf course owner, be it the developer or the association, definitely needs to be put on written notice as to your ongoing problem. I would expect that a written request for a meeting with the golf course owner would be promptly acted upon by that owner. A meeting with the golf course owner would be the appropriate way start the process towards resolution.
Sincerely, Michael Hagen