Pipes are not a safer __________ to cigarettes because, though pipe smokers do not inhale,
they are still __________ higher rates of lung and mouth cancers than non-smokers.

The sentence intends to convey that pipes are not safer than cigarettes since pipe smokers are exposed to a heightened risk of acquiring lung and mouth cancer. 

Let us try to fill the second blank first.

Pipe smokers are not responsible for higher rates of cancer. Therefore, we can eliminate option B.
'Likely to' cannot be used to fill the blank since the sentence will lack structure. If 'likely to' is used, then some other word such as 'acquire' should be used after the blank. We can eliminate option A as well. 
'Involved with cancer' is an incorrect expression. Therefore, we can eliminate option D as well. 

Option C captures both the points that pipes are not an alternative to cigarettes since pipe smokers are also subject to a higher risk of cancer than non-smokers. Therefore, option C is the right answer.