You really should employ locking tuners, but if you don't want to, try these things first. You will need some Super Lube Teflon oil for this:
1- At least 4-6 string wraps around each tuner post.
2- A drop of Super Lube oil in each nut slot (string contact/friction point)
3- A drop of Super Lube on the top of each saddle (string contact/friction point)
4- While not strung up, lube the moving arm axle points
5- Replace the old trem spring, as it might have collapsed and has lost resiliency. Tuning stability all depends on this one spring, which counterbalances all of the string tension.
If your strings are well-anchored at the posts, and all string and other friction points are lubed to minimize drag, you have a better chance of staying in tune.