The option I like best is a bit risky and involves more work, but I like the result. Take the whole thing apart and carefully bend the metal base plate to put more pressure on the outsides of the plate. (the risk is that you will bend it too much, just be very careful and check frequently. It should be bent enough to contact on the outsides of the plate just slightly more than in the middle, so when you tighten it the outsides maintain pressure) It doesn't have to be perfect, you shouldn't be flooding the base of the plate with water that often, close should count. Putty will work well, either instead of that option or along with it. Just be generous with the putty, both above and below the plastic plate, and then scrape away the excess. Plumbers putty won't solve every problem, but it usually won't hurt anything.
Avoid the silicone. It won't do any better preventing the leak, but it will cause absolute nightmares for the person trying to replace the faucet in the future.