To have loose lips means to have a big mouth, susceptible to talking about everything and everyone. Sinking ships refers to anything from small acquaintances to long and hearty relationships (with friends or a significant other). So when one says loose lips sink ships, one is basically saying if you can't shut up you are going to end hurting people, usually psychologically or emotionally. Loose lips sink ships comes from World War I and/or WWII, when sailors on leave from their ships might talk about what ship they sailed on or where it had come from, or where it was going. If they talked too much (had 'loose lips') they might accidentally provide the enemy with anecdotal information that might later cause their ship to be tracked, and bombed and sunk, hence 'Loose lips sink ships.' Later, it came to mean any excessive talk might sabotage a project.