In most situations, money is the better bet unless you are on the verge of winning or losing some colonies to rebellion and is desperate for influence. An even better way is to make them pay a tribute instead of a lump sum money as then you'll avoid this issue:
Keep your treasury below 20,000 credits. When the treasury goes over 20,000 credits, your empire gets a 20% penalty to its economy.
Be wary when trading technology to other races, especially if you are the only one with the technology. They may turn around and trade it to other civilizations for a quick profit and thus everyone will have that technology within a few turns. If you want to maximize your profit, try to be the one to sell your technology to everyone instead.
Finally, the influence you gain does affect tourism. There is an overall galactic tourism income amount, and your share of the pie depends on how much influence you have when compared to every other civilization in the galaxy. The income you gain is a strict ratio:
your income = (your influence/total influence)*(total tourism income)