I haven't been there since 2007, so I don't know how valid the following is.
The beach near the Rwy 09 end on the north (Kuta) side has been fenced off. The beach on the south side doesn't get you close enough for photography.
There's a dirt road along much of the runway on the south side. There's a high fence, and some crash gates with bars that you can shoot through. One or two knolls that you can stand on may allow some shots of aircraft taking off. The spots are not great and the airport patrol in their blue pickup truck tens to send you off if they see you. It may help if you look Iike a local.
Near the Rwy 27 end I took some nice shots from someone's roof through a friend, but the roof was so flimsy that the owner won't want to repeat this I think. However Roberto Prawiro seems to have used a very similar spot, I would drop him a message if I were you. There's also temple there that might make a good spot, but it doubt it will welcome spotters.
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Photo © Roberto Prawiro
Some people stand on the narrow main road on the Rwy 27 end, but I think it's dangerous and unpleasant.
There are some photos taken from the north side near the crossroads with Jalan Raya Tuban and the statue of Lt Col Ngurah Rai. It probably involves climbing on a little wall somewhere. There's a traffic police post nearby though.
Finally, some people have bribed their way airside.
To get around it's best to hire a scooter if you're staying in Kuta.
[Edited 2013-05-25 15:28:53]