My house is down the street.
Does the adverb down modify is, or street?
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Down here is not an adverb but a preposition, heading the prepositional phrase down the street.
The PP as a whole acts as — well, now it gets tricky. The grammar I grew up with would have called it an adverb of location modifying is:
Where is the house? There. It's down the street.
But the older I get the less I like that. I'm inclined to say that the PP acts as an adjective of location modifying (actually, “predicated of”) house:
Which house? That house. The house down the street.
Which all goes to show how much more flexible the language is than the categories we use to describe it.
But in real life it's whatever your English teacher says it is.
Adverbs cannot ever be used with copulas, as the copula essentially has no meaning and can be omitted, as it is in Russian and AAVE. A russian proverb: "Sex without love -- work." Usually the word "is" is used in the translation to English. Therefore the prepositional phrase is an adjective, as copulas are followed by a noun in the nominative, not the accusative, or by a descriptive adjective which directly modifies the subject of the verb, e.g., the sentence "John is green with envy and hates his brother now." could just as easily be "Green with envy, John hates his brother now.". Green modifies John. Same deal with "down the street".
You cannot "be" swimmingly, but you sure can "get along" swimmingly.
So to answer the question, the old-fashion way I was taught:
"Down" is not an adverb, and it does not modify anything. "Down the street" is a prepositional phrase used as an adjective, and it modifies "house". For example, look at "upon", which is never an adverb. My house is "upon the street" yonder. You can say someone sat "upon the witness seat", and that he's "going down" and end the sentence right there, meaning he's going to prison because of his crime, but you cannot say he's "sitting upon" unless you say upon what he is sitting. These phrases give the location, not destination, as in "going down to the 7-Eleven", an adverbial prepositional phrase. People speak of going "up to Denver", because it is north of Santa Fe, New Mexico, but also "down to Denver", because it is at a low altitude, only 5000 feet. In French Canada, everywhere is "l`a-bas" = "down there", and everything is "l`a", that is, "over there", even when immediately close at hand. I understand the chicken lips would say "ici", that is "y" = "there" plus "-ci" = here. People also talk of going down to places that are less important, up to places that are more important, or just use "down" to mean "over to". In Chamisal, New Mexico, people go "into town" or "up town" meaning north to Taos, or "downtown" meaning the much bigger city of Santa Fe, which is both south and larger and possessing a central area called "downtown". Taos does not have a downtown, because once you leave the central area, you are in open fields.
Rivers are the exception.
No one goes down the Nile and ends up in Ethiopia, even if it is south of Alexandria and reachable by water.
"down the street" is a preposition group that is typically used after verbs of movement as in
Sometimes such preposition groups can be used after the linking verb to be as in
I think one has to supplement "Greenwich is at a point that you reach when you go down the river" to understand the development of this use.