On one level, this question is overly difficult to answer, because professional philosophers (not sure why you've put “professional” in shudder-quotes?) are not a unified bunch, and worse, respect can mean a lot of different things. However, on another level, there is a way to answer this question that applies to any thinker.
You can find a first kind answer by checking whether his work is well-cited in the professional philosophy literature. That is, do philosophers find his work worth engaging in their own work? Has his work been treated as part of the professional conversation among philosophers?
And second, though it would take much more time, you could glance and see whether at least some those mentions are generally positive, and most say something other than “it's rubbish.” Philosophers routinely criticize philosophers they respect and declare them wrong, so only some kinds of criticism signal lack of respect, and the existence of citations is a good measure of respect. But philosophers also occasionally also cite publications as part of arguing that certain positions or arguments are rubbish and not respectable, so the existence of citations is not a perfect measure.
Towards at least the first kind of answer, you could look at Alain de Botton's citations on Google Scholar. This search reveals that he is fairly well cited, with publications in the three digits. However, when you click on the citation links there, you should check whether they venues include philosophy journals or what are clearly philosophy books.
I haven't done a more detailed search for you, but here's a suggestive starting point: At this point (February 2014), among the top 20 citations for his top 20 most-cited works, I don't find any philosophy publications listed as citing those works of his. However, you would need to dig deeper to get a fuller answer.