There are scores of dockage options behind private homes throughout Lauderdale and the surrounding area, which will be your most affordable option. As andy mentioned, some will have different restrictions than others, but if you're not looking to live aboard, most any will probably be suitable. Craigslist is one source, another good place to start would be the classifieds in the WATERFRONT TIMES...
Waterfront Times
One of the more affordable real marinas in the area, is the Las Olas Docks, owned by the city of Ft Lauderdale. Whenever I stop in Lauderdale with my own boat, that's where I'll go if staying in a marina... Nice little place, the manager Levend is a great guy, and the location can't be beat... That will likely be one of your biggest challenges in finding an affordable location, is finding one that has convenient ocean access. Many locations behind private homes are likely to entail a LONG ride to an inlet, dealing with several bridges, before you can make it outside. Las Olas is a great location in that regard, not too long a ride down to Port Everglades, with only the Las Olas Bridge for you to deal with...
City of Fort Lauderdale - Las Olas Docks
Hate to break it to you, but that area IMHO has to be one of the crappiest locations on the entire East coast to keep a sailboat. It's no wonder that even on the most perfect of days, one will rarely see more than a handful of boats sailing off the beach between Jupiter and Lauderdale. Jupiter and Boca Raton Inlets can be very dicey for a sailboat, and Hillsboro isn't much better...