Dataset on university enrollment? I'm looking for a dataset with stats on tertiary education enrollment in the US, per state, at least a couple decades back. I've searched but cannot seem to find anything. Could someone help me out?
The National Center for Education Statistics , Digest of Education Statistics does have post-secondary education (university and community college) enrollment in the US per state from 1995 to 2017. There are some tables that has the data going back to 1970 but it skips some years so you may want to perform a  some sort of missing data imputation method (i.e. linear interpolation) to impute the middle years,  if you need all the years for your analysis.
However, their website does require some skillful navigation to say the least.
Hopefully,  my instructions will help.  Once you go to this website: https://nces.ed.gov/programs/digest/
On the left pane of your window go to the List of tables and figures section.  There is a  dropdown box below, select a year.
It goes to 1995.  Once you have selected a year go to chapter 3  Postsecondary Education, then select degree-granting institutions, then enrollment.   At this point you will need to find the table that has the state data.
From what I understand of the data that you are looking for, I saw it is in table 195 for most years (1995- 2006) , then its table 190 in 2007,  2008- its table 203  , 2009 - table 219 , going forward for most years the table number that has the state data your looking for seems to change but I am pretty sure its there in chapter 3.
For your research project a recommendation is to pay attention to how the NCES Digest of Education Statistics changes how define their state level enrollment data.   They also have more education data by state if you are curious to see what they have that could support your work.
If you run into trouble with these tables I would recommend to call their office.  The staff list is located here https://nces.ed.gov/ncestaff/stafflist.asp
I hope this helps.