Let's start with standard GB theta theory. In your sentence "I saw the tall kid", "the tall kid" is an NP,1 with "kid" as the head. There is an AP here, as you suggest, but it consists of nothing but the head A "tall", and it's attached as an adjunct to the N. Something like this:
Adjuncts don't assign theta roles to their heads. Theta assignment is the mapping of roles from a theta grid onto argument position in the tree, and adjuncts don't have arguments.2
Put another way, from closer to the semantic side of things, the way "tall" functions here is attributively, not predicatively. Attributive modifiers don't assign theta roles. So, even if "tall" does have a theta grid, there's no theta assignment by "tall" here.3 Therefore, no double assignment to "kid", and therefore no violation of the theta criterion.
Now, from your comment:
This sentence is taken from Naoli (1989) 'Predication Theory: a case study for indexing theory', p. 76, she assumes that there's an embedded predicate in this sentence with a lexical head ADJ.
So apparently, Napoli is arguing for some theory where attributive modification is actually predication in disguise—not just at the semantic level (as many have argued), but syntactically. We really have a syntactic clause, built out of an AP instead of a VP. I don't know if that CP is instead of the NP, or inside the NP, or inside the N', or what, but however you work it out, presumably "tall" ends up in a position to assign a theta role to "kid".4
But if there's really an A-headed clause with "tall" as the head, then "kid" is no longer the head of an NP in VP-complement position in the outer sentence. So "see" can't assign a theta role to "kid", so there is no double assignment.
But then where does the theta role assigned by "see" go? An unassigned theta-role would be a different theta criterion violation, but it's just as bad.
That's hard to answer without some picture of how Naoli analyzes "the tall kid" here. I could invent my own structure and come up with an argument for it, but the chances that it will be the same as Napoli's are pretty slim, and therefore it isn't likely to help you at all.
But in general terms, just compare it with "I saw that the kid was tall": see has [CP that the kid was tall] as an argument, "kid" doesn't get theta-marked by "see", and there's no theta criterion violation. So, if "I saw the tall kid" has a covert CP either as or inside the argument for "see", it's essentially the same, even if the details are different.
Or, if you look at things from a phrase-marking rather than head-marking perspective: "see" can assign a theta role to the CP or NP-with-a-CP-in or whatever [the tall kid]. But that isn't a problem, because "tall" isn't assigning a theta role to [the tall kid], it's assigning a theta role to [kid], an NP deep inside that CP.
Again, it's hard to get into details without knowing Napoli's theory, but again, compare any other case with an embedded clause: [I saw [NP the boy [CP that ate the chips]]. The fact that "see" marks [the boy that ate the chips] while "eat" also marks [the chips] isn't a theta criterion violation just because [the chips] happens to be inside [the boy that ate the chips].
1. Or it's a DP, with "the" as the head, and various things get more complicated, but the end result is the same, so let's ignore that here.
2. Well, they can have arguments internally, as in "the taller-than-average kid". But let's not get into that. They don't have external arguments: "kid" is not an argument of "tall"; "tall" is an adjunct of "kid".
3. In the original version of theta theory, only verbs had theta grids; "tall" has no theta roles to assign, so there's no issue at all. That causes problems with, e.g., Japanese "Jon wa segatakai" ("John topic-particle tall-adjective-as-predicate": "John is tall"), where "segatakai" is an adjective, which moves to I position to head the sentence, so presumably it has to be the thing assigning a theta role to the subject (well, topic, but forget that). So then maybe even in English "John is tall", the adjective "tall" is assigning the theta role through the copula in some way possibly involving lots of complicated movement. Under such theories, theta grids are only expressed by predicates in a "relational head position", like head of VP or IP, rather than being expressed by anything that has them. So "tall" does have a theta grid, but it's ignored when "tall" is the head of an AP that adjoins to an N', because it's never in a relational head position.
4. For example, maybe it's essentially the same way "segatakai" theta-assigns in the previous footnote.