I have a set a pipes that makes a loud vibrating sound. It comes on when the hot water is turned on in one of the upstairs sinks, showers or tubs. When this happens, the basement bathroom sink starts to leak. If either hot or cold water is turned half way on, in the basement bathroom sink it will stop. However, if the upstairs water is still running, the noise will resume as soon as basement sink is turned off. The vibration sound is coming from the pipes behind the basement bathroom. I have read that you can stop the vibrating sound, by anchoring the vibrating pipes securely, which I can do. However, is there a bigger issue here, then just the vibration of the pipes. I would hate to mask a serious problem and end up with a really big problem. The sounds started about 1 week ago.

Vibration of pipes when water is running is usually a sign of a loose washer in a valve. This vibration can vary depending on water volume (therefore the speed). If this occurs only when hot water is used the defective valve would be in the hot water line. Opening the basement sink faucet would change the volume only in the main line out of the water heater. It would also change the volume of the cold water line into the water heater. Does the water heater have heat trap nipples. These are a from of valve. Try changing the setting of the cold water stop valve.
Feel the pipes for vibration to locate source.