Many people keep a locket closet (make sure it is thoroughly locked with pins going into the cement and a strong lock) to discourage "finders of your things." Even in the nicer developments, the rentals often have these. If you have smaller items, I know PriceSmart in SD has safes for sale right now.
If you REALLY value it, leave it in the US (but that is no guarantee either as our house in the US was robbed while we were down here.) I suppose the main lesson is to have less STUFF that you care about. I wouldn't bring real silverware down.
As for the airports in general, I've found that the customs agents like chocolate. One time I declared the cases of Hersheys and Snickers I had for my chocoholic kids and dh, and they had me open it up and wanted me to pay a ridiculous tax OR give them one of the boxes (they started wanting half of them so I figure I got off okay.)
When I am bringing things down for the orphanage, I never get hassled--however, others I know have. There have been many people bringing coloring books, candy, toothbrushes, etc. where 1/4-1/3 of the totals (for 160 kids) have gone "missing."