Yeah, I'm afraid that's been the case at my alma mater more often lately due to budgetary issues...Kind of a pity we can't even afford to host prospective, accepted recruits when they come from out of state/country.
What I've seen done is for the department staff to coordinate with the department's student representatives in organizing the whole recruitment visitation event, which has included making hosting arrangements for visiting recruits. Usually it has involved sending emails around to current grad students asking if anyone can host a recruit. Responses and outcomes have been pretty good from what I've seen. I think that's something of a small miracle, considering most grad students live rather modestly. I don't think I was ever in a position to host anyone, but it seems enough other students have managed to make it work that the department can get away with not supporting visitors' lodging needs financially.
I'd expect faculty to have somewhat better accommodations available, judging from the few whose houses I've visited – much more spacious than any grad student's I've ever known. I'd also expect faculty to have a more vested interest in cajoling good recruits into enrolling, and also more interest in getting to know these students ahead of time. Faculty might have longer histories in the area as well, and therefore make better hosts and offer better introductions to the city outside the university. Some faculty are very busy, but some are tenured (insert joke about sea cucumbers here, but the implication that tenured professors aren't busy isn't to be taken too seriously), and no grad student has this luxury – they're all busy, or at least should be. For these reasons, I'm inclined to argue that faculty would actually make better hosts than grad students overall, if they could just be convinced to take that extra responsibility.
However, I'm not sure a faculty member should host his/her own recruit. Personalities can clash in unpredictable ways when people share their personal spaces with one another, and a long-term advisor-student relationship is not one with which to take such risks lightly. People sometimes depend on a sort of professional distance in relationships with authority differences (i.e., superior-subordinate relationships, to whatever extent that fits the advisor-student scenario), and learn to regret lowering their personal boundaries through particularly bad experiences. These are easier to put behind oneself if one doesn't have to work with the other person continuously over several years thereafter.
Furthermore, one wouldn't want to create an implicit expectation in a department that faculty should host their own recruits. Many people choose not to live in tidy, spacious, or peaceful places that a visitor of unknown sensitivities might need just to be able to relax and sleep. This is why it's important to request volunteers for informal hosting if it can't be done through a professional hosting service such as a hotel (or even a hostel). Some people are reluctant to host others for very good reasons, and shouldn't suffer others' judgment for failing to fulfill their expectations when they would be unreasonable. It's all too easy to produce a culture of implicit obligation inadvertently like this, so this justifies preventative precautions.
@ff524's points are well-made on this matter too, and the dorm suggestion isn't a bad idea either, but from a recruit's perspective, I would prefer to be offered a room in a grad student's house / apartment / foxhole. Dorms tend to be dominated by undergrad culture, which at some universities won't make the best impression on recruits. There's enrollment-incentivizing value in being welcomed into a tight-knit, professional environment that gives a realistic impression of what life is like for other grad students, who presumably don't live in dorms (unless they're RAs). Staying with other grad students is a chance to talk about the university or the city over breakfast or dinner, whereas undergrads might make somewhat less useful company.
I'm not sure whether I'm giving undergrads too much credit or too little in saying no more or less than this...Some dorms and undergrads could make very nice environments I'm sure, but others could be downright awful. It seems to me that there's much more risk involved in the dorm proposition than in having other grad students host recruits.