This is a good question. I've not thought about it whatsoever so in my infinite off the cuff wisdom here we go.
I could see an argument for "no". Libertarians do observe borders and there is no claim to be allowed to come in and enjoy what we have built. In other words, justice is not violated if we prevent people from seeking asylum.
The problem comes in when people do seek asylum and you do want to do something about it (let us suppose a majority agree). And it's a much bigger problem than taxes but I think we should constrain the topic there. The question is, is it right for the government to levy a tax to hear the asylum cases. I think the libertarian answer is no. Perhaps the government could let one opt out of such a tax if they were against it or if they felt the government was doing a bad job of it etc. The funding and influx of people would track with what people cared about enough to put their money toward. This should also be the case for a lot of other things the government puts their fingers in. Is it really a concern for the people or do you just have pet project you think is really important? If it is really a concern you can measure it by how much money people will put behind it. (that is at least the measure which forces people to have "skin in the game".)
As far as the exact quote goes
Everyone has the right to seek and to enjoy in other countries asylum from persecution
I agree with the first part, they do have the right to seek. (we should probably discuss rights at some point) Enjoy implies they also have the right to enter whichever country and participate. No, that is not a right. That is a privilege an asylum seeker hopes to attain.
What do I think? I think most people would agree with funding an asylum claim facility so there's probably little danger letting them voluntarily give to such an effort. Their dissatisfaction would also be known when funding dries up. Nice message to the government that they are doing a shoddy job in your opinion.
Now that I'm thinking about it. We could have a basic keep the lights on tax that funds bare bones stuff and perhaps skeleton crews of other things. Everything else would be funded by the donations of the people. It would then behoove the state to make sure everyone was as rich as possible and as generous as possible. The other major upside is that the state would have to fight for funding through performance rather than need. The people could decide to voice their displeasure in something by withdrawing their funding and the government would actually feel it. The people would matter and they would have incentive to pay attention to what government is doing.
I don't have an immediate aversion to the idea. Does it check out with you?
The argument would then, obviously, lead to what is the bare bones. Yes that is where we are now. Everything is the bare bones. That would have to be more clearly specified. I'm thinking military/borders/country security up to xxx dollars, and courts/property rights/police up to xxx dollars (and probably certain laws should be considered bare bones laws).
Basically, let's build the country that every asylum seeker wants to come to.