Should I ask for a recommendation letter from a professor who has a bad non-academic reputation? I am an undergraduate interested in applying for graduate school in a physical sciences program. I have a good relationship with a few of my professors (it's a very small department). I would like to ask if they would be willing to write a letter of recommendation for my application.

There is one professor who I respect greatly and who has told me in the past that he would be willing to write a glowing letter. He has a great academic reputation; he publishes the most out of all the professors in the small department and in (what I have heard) are top journals in his field.

The trouble is that he has a poor non-academic reputation. For instance, a quick Google of his name will yield a page full of news articles regarding his troubles with the law. These are the top results from the search.

Would a poor public reputation like this have a negative impact on the effectiveness of the reference? In theory, I know it shouldn't. Yet I am aware that real life hardly ever works as 'in theory'.

EDIT: the "troubles with the law" involve an impaired driving conviction. For context, this is in North America.
To me, the key question seems to be whether the type of misbehavior is something that is likely to be seen as impinging on their judgement as an academic.  

For example, if the person was known to have been prosecuted for fraud in a company they ran, that would be quite likely to affect perception of their honesty and judgement in recommendation letters.  In the example you give, however, I think that an impaired driving conviction is likely to be viewed as a form of misbehavior largely orthogonal to conduct and judgement as an academic, and therefore not likely to be any problem for you.