Short answer: To be welcoming probably means to
Be patient and polite when explaining the scope of the website to those who aren't very familiar with the scope of MO.
Give newer members an opportunity to get involved with voting to close questions. Having only 6-7 names appearing in every closed question can be intimidating.
Encourage professional behavior in order not to turn away those that are familiar with the scope of MO.
Whether Mathoverflow is perceived as a welcoming community depends on the observer. If it is someone who is not very familiar with the website, then I think that being welcoming boils down to politeness. This is not just about redirecting questions that may not be research level, but also explaining various other procedures such as other members of the community editing one's posts, changing the tags, closing as duplicate etc.
There are aspects of the website where the community's consensus is constantly evolving. When it comes to deciding what constitutes a "research level" topic, some users have a higher bar than others. As a corollary, the experience of a new contributor will be shaped differently depending on whom they come into contact with. For example, I have noticed that user abx has been vocal about casting votes to close on algebraic geometry questions that others might find ok or borderline, you may even have experienced this in a recent question of yours. I don't think that there is anything wrong with this (in fact I believe that many topics on MO could benefit from "raising the bar" a little bit) but perhaps it would be helpful if a new user could tell the difference between the opinion of a handful of users and the community as a whole.
One way of achieving this is to discourage users from identifying themselves with all of MO. In the context of borderline questions, using phrases like "I think that your question is covered in a standard textbook like X and might have a better chance at attracting more elaborate answers at website Y" is probably better than an absolute "your question is not acceptable/welcome/appropriate on MO". They essentially achieve the same goal, but the perception can be different. I guess the desired perception should be more along the lines of "community moderated" and less "policed by a small set of users". The first scenario is one where every member can feel like they have a voice whereas the second is one where some members are left to feel sort of helpless.
I think that the case of borderline questions is truly the place where you can clearly see how welcoming we choose to be as a community. Perhaps if we were able to treat them with more dialogue we might start seeing a change in perception. Here is what I think is a great example, where instead of getting closed, a question received constructive criticism as well as an edit that improved it.
Now, for the case of someone who is familiar with the scope of MO, there are other things that matter more. The impression I get form personal conversations is that some of the factors which may paint an unwelcoming picture of MO include: 1) Non-mathematical debates that turn offensive or rude. 2) The lack of a unanimous agreement against hate speech/symbols (without bias on which country or culture they come from). 3) Users regularly posting controversial statements and then deleting them for the sake of creating confusion etc. Running into such threads on MO or meta can give the impression that this is not a professional environment. This can turn potential users away because they do not feel like the level of decorum is high enough to protect them from personal attacks, prejudice, bullying etc.