The problem with misplaced/dangling modifers is a logical one: the noun after the comma must be something that can logically be modifed/described by the modifier itself.
Example: Running down the street, a brick fell on his head.
the problem with the sentence above is that the noun following the modifier is "a brick", which cannot logically be running. Take the sentence as literally as possible - if the brick is the noun following the modifying clause, the the sentence seems to erroneously indicate that the brick was the on doing the running. One way to correct not necessarily the best, or only way) is to change the sentence so that the noun following the modifier is a noun that can be running:
Running down the street, he was hit in the head by a brick.
Here, the pronoun "he" is something that could be running, and thus can be modified by the preceding modifier.
(BTW, another way to correct the problem would be to leave the main clause as is, but add a subject, verb and connector to the modifier, thus turning the modifier into an independent clause which doesn't not need to modify anything:
While he was Running down the street, a brick fell on his head.)
Apply the same overall logic to your sentence, and the problem seems apparent: the admissions office itself is the noun that follows the modifier, but the office cannot be "based on a recent decline in enrollment" - the DECISION should based on the enrollment. Two possible fixes, as explained before:
1) Rearrange the sentence so that the correct noun follows the modifier:
Based on the recent decline in enrollment, the admissions office's decision was to reevaluate its recruitment strategies - awkward, but fixes the modifier problem, since the decision is a noun that can be based on declining enrollment.
2) Rearrange the modifier so that it's either an independent clause, or a modifier that could logically modify the admissions office:
Because of the recent decline in enrollment, the admissions office decided....
"A because of A" type clause is still not an independent clause (it has no subject and verb, per se), but at least it's not illogically hinting that the office itself is based on the recent decline.
Hope this clarifies things a bit.
