Perhaps you have not completed the required mission goal yet. To get there you must get to Julie Langford's office and get the audio diary "Lazarus Vector" from the safe behind the botanical print. The code for the safe is 9457, the same number Langford scrawls on the office window as she dies. After listening to the diary your new objective will be added, just follow the arrow to the Farmer's Market.
I've solved it this way:
1. Go to the locked door at the Bathysphere Station on the map
2. Use the cheat code "Ghost" to go through the locked door then use "Walk" to cancel out "Ghost" (see how to use cheats here http://bioshock.wikia.com/wiki/Code)
3. Use the elevator and go to Fort Frolic.
4. Now immediately after you reach Fort Frolic go back to Arcadia and now the door to the Farmer's Market should be opened.
PS: if the Bathysphere Station's door is locked again then use "Ghost" to go through it and then use "Walk" right after you get out.