We've never seen Alfred act on its own volition to imprison someone. At the times when the Denarians were on the island, there was no Warden to give the order to imprison them. Almost all the creatures and beings in that prison are there because someone (i.e. the White Council I'm pretty sure) put them there.
Could the prison hold a Denarian? I imagine so. Could it hold a Coin? I'm not so sure. If nothing else, there would be the risk of infernal intervention.
I have to agree with Mira on this. Maybe Alfred doesn't like visitors on the island or maybe not, we just don't know. Apparently someone was able to operate a cannery there for a little while at least, so it's certainly possible to visit it. Alfred might not like Rashid, but Rashid still visited the island.
So I would imagine that the Council could send a party of Wizards to check the place out every so often.
And even if that isn't a viable option, it would not be that hard to keep track of who visited. The Council has vast resources of money. They could put a few boats on Lake Michigan in the area, disguised as ordinary fishermen or pleasure boaters or whatever to keep track of the waters around the island, they could hire someone to run a radar set on such boats to watch the skies about. It just wouldn't be that hard for the Council to keep track of comings and goings, at least in the real world. We don't know how hard it would be to access Demonreach through the Nevernever, but Nicodemus seemed to think it would be...difficult. And I suspect the White Council has the resources to keep track of the Nevernever near Demonreach, too.
It really is strange that the Council doesn't seem to be doing that, considering what that place is. Maybe, maybe, they're afraid of calling attention to it by watching it too closely...but I don't really buy that.
Mira has a point, this is really strange.
I suspect the Council usually does choose the Warden, when there is one. They might have to be careful to choose someone acceptable to Alfred, but I doubt that's an insuperable barrier.
Everything to do with Demonreach is a retcon it seems though.
In
Small Favor, where we are introduced to it, a large group of non-wizards were able to reside there long enough to build a cannery and dock. How? The psychic pressure of the island gave Karrin, Michael and even Thomas horrible nightmares after just a day or so of exposure. Its vibes are strong enough to get it excluded from
all maps of Lake Michigan and flight paths
into Chicago steer clear of it. That's a pretty strong ward.
In fact, the strength of that ward and the Ley lines coming from the island ought to draw magical talents like flies and mosquitos to a bug zapper. Practitioners should get a "bad idea" warning when they think of approaching, but there's no way it'd be unknown, if only for the huge ley lines coming from the center of a large body of water.
And why aren't there any older wizards using those ley lines along the shores of Lake Michigan? You'd think it would be a wizardly lake town and retreat. If you say the White Council stops those wizards because they are suspicious, why don't they have permanent presence there?
The only theory I can come up with is that it is part of Rashiid's duties as the Gatekeeper. He does know a Way to the island and has some history with it. Perhaps he pops in from time to time to see if it has a Warden. His foresight may even have shown him Harry in the role and doing a good job.