People say it both ways, although singular agreement is more common. To a certain extent, we might suggest it depends on how you're conceptualizing it:
If you conceptualize a pair of X as a single unit to which both things belong, then it takes singular agreement.
okThere's a pair of pants on the nightstand.
??There are a pair of pants on the nightstand.
We conceptualize a pair of pants as a single unit.
If you're focusing on the following noun, thinking of it as plural, then the string a pair of doesn't determine agreement, it just tells you how many of the thing there are:
okThere is a pair of kidneys in the human body.
okThere are a pair of kidneys in the human body.
Here we can use are, although we don't have to. There isn't much difference between the two conceptualizations in this case.
When a pair of X takes plural agreement, we can say it's been grammaticalized as a kind of quantifier:
If, to untrained eyes and ears, the two groups are virtually indistinguishable, there are a pair of good reasons for this. (Big Poppa's Bubble Gum Machine, TIME 1999-02-01, via COCA)
I picked a real example from a corpus to show that people really do write things like this. It's not even particularly uncommon, although it's less common than the alternative.
In any case, in this example a pair of is used much like a couple of or two:
There are a pair of good reasons for this.
There are a couple of good reasons for this.
There are two good reasons for this.
The following noun reasons appears to be the head noun; a pair of (or a couple of or two) tells us how many reasons there are. The verb agrees with the head noun, which is plural, not with this quantifying phrase.
More often, pair is treated as the head noun, and pair is singular, so the phrase takes singular agreement:
A set of gray coveralls hung from a hook on the door, and there was a pair of moccasins on the floor. (In Spare, Analog SF&F April 2004, via COCA)
Here we have the singular agreement we expect. Again, this is a real example from a corpus, and we can find many more like it.
I hope the answer has been useful so far. But I've simplified things in two major ways. I'd like to talk about the first one now, so if you're interested, read on:
One big complication is this: Your example has an existential construction with there, and in this construction the verb doesn't necessarily agree with the post-verbal noun phrase. More and more in today's speech, younger speakers are using contracted there's as an invariant form before both singular and plural noun phrases. That means many speakers would produce and accept examples like the following:
okThere's three people at the door.
But can't use there is here, only there are:
*There is three people at the door.
okThere are three people at the door.
Strange, right? We'd expect there's only when there is is grammatical. But as the English language changes, the two are becoming grammatically different from one another. There's often works when there is does not, and this sort of utterance is only becoming more frequent as time passes.
Now, if we look at the non-existential versions, we find 's only works when is works:
*Three people's at the door.
*Three people is at the door.
okThree people are at the door.
As you can see, is doesn't work, and neither does the contracted 's.
The form with there's is colloquial but extremely common among speakers of Standard English. So even if you determine the post-verbal noun phrase would take plural agreement on its own, you may also use the invariant form there's when you place it in an existential construction, as long as you're aware that it's a colloquialism and don't try to uncontract it to there is.
Which noun is the head noun, really?
The other simplification I made is in assuming pair isn't the head noun when the verb doesn't appear to agree with it. I made this simplification because I don't think the competing analysis is particularly useful pedagogically, but if you're interested in the nitty-gritty details, you should be aware that linguists have arguments for a competing analysis as well.
In The Cambridge Grammar of the English Language, the authors argue that pair is always the head noun, even when it apparently doesn't determine agreement. One way to account for this is by saying pair is 'number transparent', absorbing the grammatical number of the complement of the of-phrase. This has some theoretical advantages; after all, the head of a phrase generally can't be omitted, and the of-phrase often can with this sort of phrase!
Really, the reason this is so complicated is that grammaticalization is a very slow, gradual process, and a pair of is only partway through the process of becoming a quantifier. So really, it has grammatical attributes of both its original grammar (as the authors of CGEL would point out), but it also has attributes that make the other analysis attractive. I'm intentionally glossing over these details here, as I'm not sure they have much pedagogical value.