How far from basalt bedrock can magnetometic survey find archaeological features?
You can get a good idea of the magnitude and wavelengths of typical archaeological magnetic anomalies with a literature search. For instance this page shows a magnetic map of fire hearths http://en.wikipedia.org/wiki/Magnetic_survey_(archaeology)

Geology is a lot more complicated though and there are many variables that might affect the background magnetic signature. It doesn't matter how much you study up on basalt rock properties you still won't know enough about the rocks in your project area to predict the outcome of a survey. 

Your best hope is to use prior experience from other surveys close to your project area. If a survey worked in one farmer's field then it will probably work in the adjacent field as well. 

If no one has collected any magnetic data close to your project area then go ask the geophysicists if they will do a test line or two. If it works great, if not then try something else like magnetic gradiometry, resistivity or ground probing radar. 

Sorry I can't give you a definitive answer to your question. I understand what you are asking but geology is just too complex to give you a useful rule of thumb.