How little strength should I have in bid suits for a negative double?
First the general points - if you and partner have a fit, opponents are almost guaranteed to have a fit also. Values in the opponents' suit detract from the overall playing strength of your hand, and should be evaluated accordingly, but that does not relieve you from the obligation to share information with partner.

The scoring table clearly favours major suits over minors, and generally it is true that with both a 4-4 fit and either a 5-3 fit or 5-4 second fit, the 4-4 fit plays on average almost a trick better. Of course in a minor you must contract for a full additional trick for game, and often for the privilege of outbidding opponents. This analysis leads the following guidelines:


Don't pass if you have the values and support required for any type of raise. Your known fit improves the opponents' expected fit at about 75%, and you must inform partner of this valuable datum.
Prefer to look for a 4-4 major suit fit in preference to raising partner's known major or minor suit unless you have 5-card support for partner's major. 
Prefer to raise partner's major in preference to looking for a 4-4 minor suit fit.
Don't pass in expectation of passing a reopening double by partner unless you expect a doubled penalty greater than your game (500 white, 800 red) - if you can't yet set the opponents this much you owe partner a bid first. Bridge is a partnership game.
After the auction 1C 1D a negative double shows at least 4-4 in the majors - otherwise just bid one; such a response at the one-level should not require a 5-card suit or you are letting the opponents steal bidding room too cheaply. 
After the auction 1D 2C a negative double shows at least one 4-card major, to protect against partner having one also, playing 5-card majors.
In other low-level auctions you are promising at least 4-3 in the unbid suits, or extra values and an inability to bid any of notrump, trap pass, or a raise.