Base is a transitive verb: base something on something. The sentence is essentially equivalent to “Many companies base their judgement of students on their teachers' references”. Cast in the passive voice, this becomes (decidedly unidiomatically) “The judgement of students by many companies is based on their teachers' references”. We can expand this to “Many companies judge students in a certain way; that way is based on their teachers' references”. As this is rather clumsy, the sentence is condensed to “Many companies judge students based on their teachers' references”.
The past participle is used as an adjective. Its meaning is related to a the use of the past participle in the passive voice: it qualifies what is based (the judgement), not who is basing (the companies). Here is a slightly artificial sentence that illustrates the use of a participle phrase with a present participle, with an active meaning:
Many companies evaluate students, basing their judgment on their teachers' references.