Are seasons capitalized when referring to a specific season? For example, is it, "in the Fall 2007 semester I provided therapy to Frank" or "in the fall 2007 semester I provided therapy to Frank?"?

In all of my experiences they are capitalized.

Fall, Spring, Summer, Winter semesters.

Seasons are only capitalized when used in a title. You are right to put it as Fall semester 2007.

AstroChuck is absolutely correct.

“Don’t capitalize the names of seasons when they are used generally (capitalize seasons when they are used with another noun and thus function as proper nouns):

spring fall Spring term”

Source

It’s like “Mom said you can have a cookie,” as opposed to “One kid’s mom from our class made some cookies.”