Yes. You still control the trigger so you are the one who will end up losing 2 life. Giving it away (or destroying it) will not affect the trigger in any way.
603.3a. A triggered ability is controlled by the player who controlled its source at the time it triggered, unless it's a delayed triggered ability. To determine the controller of a delayed triggered ability, see rules 603.7d-f.
upkeep triggers, 2 damage on the stack
in response i donate the idol to my opponent with bazaar trader
do i still take 2 damage?
Yes. Here's the sequence:
Your upkeep step begins. This triggers the ability of Jinxed Idol.
You would get priority, but there is a triggered ability to put on the stack. You put "At the beginning of your upkeep, Jinxed Idol deals 2 damage to you." on the stack. Note that this ability is now a separate object from Jinxed Idol; even destroying or otherwise removing that artifact won't remove this ability.
You get priority and activate the "T: Target player gains control of target artifact, creature, or land you control." ability of Bazaar Trader. For the targets, you choose your opponent and your Jinxed Idol, respectively. You pay the cost by tapping Bazaar Trader.
You get priority and pass.
Opponent gets priority and passes.
The top object on the stack (the ability activated in step 3) resolves. Opponent gains control of Jinxed Idol.
You get priority and pass.
Opponent gets priority and passes.
The top object on the stack (the ability put there in step 2) resolves. Jinxed Idol deals 2 damage to you.
You get priority and ....
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Why bother with mere rulings when so many answers can be found in the Rules?
if i have a jinxed idol...
upkeep triggers, 2 damage on the stack
in response i donate the idol to my opponent with bazaar trader
do i still take 2 damage?
Yes. Here's the sequence: