I'm playing through the GOTY version, and I can access all the DLC except for Broken Steel.  Do I need to finish the game before accessing it?

When you complete the original game it will show a movie showing what you did in your playthrough. Normally at this point the game would end and you would go back to the main menu but instead it will go back into the world at the start of the Broken Steel content.

I just finished it myself.  A very good episode of DLC.  I went back and played the game but as stated above you have to finish the game first but you can go and do the side missions with it as well.

You don't have to have completed the main quest to take advantage of the increased level cap, though.

" You don't have to have completed the main quest to take advantage of the increased level cap, though. "