You didn't say what paints you are using, I assume enamels, which are usually contained in metal or glass with metal lids. A slight odor of solvent on opening a sealed container in which they have been stored is probably unavoidable. But a strong odor probably indicates a problem.
When you open a jar of paint, the first thing you should do is wipe the inside of the cap absolutely clean—not a trace of paint or clear resin anywhere on the gasket or threads. A paper towel dampened with a small amount of solvent, or a cotton swab, will make this easier. The second thing you do is wipe the bottle lip and threads clean. Then and only then start using the paint. Repeat this cleaning process as necessary before closing the jar, and close the jar tightly. If you have trouble closing the jar (arthritis or whatever) a jar opener, used in reverse, will help.
Keep the paints in a cool (not cold) place where they are not exposed to direct sunlight or other sources of heat, especially for long term storage.
This applies to all paints, acrylic, enamel, lacquer, oil, etc.