It appears Liberius did have some veneration in the West, but it was not universal (not sure the history in the East or when his veneration became universal there). Opinions seemed to have been mixed. While he was not included in the Roman Martyrology, he is included in Wandelbert’s (a somewhat famous one compiled by a 9th century monk, which is based on earlier ones, like St. Bede’s). The pseudo-Hieronymian Martyrology noted memorials for his deposition/departure, which is what is memorialized in the Coptic churches.
On the other hand, there is the tradition that he wasn’t so great. St. Peter Damian, in the context of defending the validity of priestly ordinations performed by simoniacal bishops, puts Liberius in a pretty bad light among examples of other bad Popes:
And so it was that all the ordinations performed by Liberius, who was both a heretic and a turbulent man, were considered valid and immutable. Liberius, moreover, who was deceived by error and disbelief, is known to have subscribed to the Arian heresy, and because of his transgressions many horrible crimes were committed. Many priests and clerics were killed because of his wickedness, and the remaining Catholics were forbidden to use not only the churches but also the baths. Subsequently Liberius apostatized and lived on for six more years. Yet whatever he did regarding ordinations remained valid and firmly established in all its vigor.
More recent scholarship has done more to confirm the more positive views of him and make them more popular in Rome. For example, he is praised in letters of Popes Pius IX and Benedict XV.