I've been told that both sentences are correct, so how can I tell next time which form to use, plural or singular?

Do either of you have any money I can borrow?

Either of the plans is equally dangerous

They're both singular. But the first is second person.

Consider:

Does either of them have any money I can borrow?
Do either of you have any money I can borrow?

That's because we say you do and he/she has.

EDITED: In fact, "Does either of you" and "Do either of you" are both used by native English speakers. See Ngram. But if the subject isn't either of you, you should use a singular verb with either of.

I agree with Peter Shor that in "Do either of you have any money I can borrow?", the use of "do" might have more to do with the presence of the second-person pronoun "you" than the presence of the word "either". To me, "Do either of them have any money I can borrow?" seems wrong, although not glaringly so (so I'm not sure what percentage of other English speakers will agree with me).*

In your question, you bolded have, so it might be worth mentioning that it is an infinitive in that sentence, and wouldn't change form based on the grammatical number of the subject. You would also use "have" in a sentence like "Does he have any money I can borrow?"

"Either of the plans is equally dangerous" sounds a bit odd to me, but not because of the "either of the ... is..." construction. For some reason, I don't like how the word "equally" is used in this sentence ("Either of the plans is dangerous" sounds fine to me). If I could, I would want to rephrase this sentence to "Both of the plans are equally dangerous" or "Both plans are equally dangerous."

*I just searched the BYU TV corpus for . DO EITHER OF THEM and got 3 results, compared to none for . DOES EITHER OF THEM, which makes me even less confident in my characterization of "do either of them" as wrong in the first paragraph.

Some of the purists will tell you that "either" should take a singular verb. But that's not always how people speak.

In the GloWbE corpus, "do either of" gets 237 hits, "does either of" gets only 30. (I have excluded cases where the following word is "you", because as Peter Shor points out, that takes "do" whether it is singular or plural).

This answer discusses the question in more detail.

Do either of you have any money I can borrow?

Either of the plans is equally dangerous

You are using two very different constructions. Let's highlight all the verbs:

Do either of you have any money I can borrow?

Either of the plans is equally dangerous

• 
  do - auxiliary verb (non-defective), only other present tense form is "does", which is only used for third-person singular


• 
  have - infinitive. We can substitute "be" and a different final clause to prove it's an infinitive: "Do either of you be good for your parents?" No finite verb (i.e. conjugated) works in that position


• 
  can - modal verb (defective)


• 
  borrow - infinitive


• 
  is - third-person singular present indicative of "to be"

In total there are three places a conjugated verb can appear in the two example sentences:

• 1st sentence main clause: Do [subject] [infinitive-verb] ... ? (subject-verb inversion due to it being a question. Do-support due to "have" being used in a non-auxiliary sense. Note: "Have either of you any money I can borrow?" with "have" being used in an auxiliary sense is perfectly acceptable, and is used frequently in some English variants.)


• 1st sentence sub-clause: [subject] [verb] ... ("I can ..." Defective modal verb "can" used which is invariable in each tense)


• 2nd sentence main clause: [subject] [verb] ... ("Either ... is ...")

In both sentences, the subject of the main clause is "either of NOUN". Here "either" is being used as a noun. If the subject was just "either", it would be a pronoun, and if it was "either NOUN" it would be a determiner. Technically, that means that the grammatical number and person of "either" should (prescriptively) determine the conjugation of the verb.

Either_PRON is equally dangerous.

Either_DET plan is equally dangerous.

Either_NOUN of the plans is equally dangerous.

Note you can't use a determiner immediately before a pronoun, so the other example sentence becomes:

Do either_PRON have any money I can borrow?

Do either_NOUN of you have any money I can borrow?

Both the pronoun and the noun are third-person singular. Determiners don't inflect for grammatical person in English because they always attach to nouns, and nouns are always third-person, and only sometimes inflect for number, e.g. "this"/"these", "that"/"those". Note that when "either" is used as a determiner, the plural is clearly ungrammatical: "either *plans".

In all cases, "either" means "one of two". This means that technically, it is always singular. Compare:

A herd_NOUNis a good choice. / It_PRONis a good choice.

One_PRONis a good choice.

Either_PRONis a good choice.

The_DET cow is a good choice.

One_DET cow is a good choice.

Either_DET cow is a good choice.

A herd_NOUN of cows is a good choice.

One_NOUN of two is a good choice.

Either_NOUN of them is a good choice.

Cows_NOUNare a good choice.

Two_PRONare a good choice. (Note: (The number) two is a good choice.)

They_PRONare a good choice.

However!

For a lot of people the singular sounds strange and they prefer to use the plural. Descriptively, you'll find in many cases that the plural is used more than the singular.

Prescriptively, only the singular is correct. Descriptively, the plural can be equally or more correct than the singular.

Either_PRONare a good choice.

Either_DET cow *are a good choice. (always incorrect)

Either_NOUN of them are a good choice.

• n-gram of Do/Does either of us (almost equal usage)


• n-gram of Do/Does either of you (strong preference for "Do either of you", about 10 times stronger than "Does either of you")


• n-gram of Do/Does either of them (almost equal usage)

Nitpick:

Either of the plans is equally dangerous

exhibits poor grammar, unless it is part of a longer sentence:

Either of the plans is equally dangerous compared to some third plan.

If it is meant to be an entire sentence, it would be better to say:

Both of the plans are equally dangerous.