Does the solvent used affect basic strength?

In [this](https://chemistry.stackexchange.com/questions/47582/nucleophilicity-basicity-in-protic-and-aprotic-solvents) question, it is explained how basicity and nucleophilicity are different and how the solvent used affects nuclephilicity.


What I wondered was whether there would be a difference in the basic strength of say $\ce{Cl-}$ in $\ce{H2O}$ (a protic solvent) and $\ce{DMSO}$ (an aprotic solvent).


My answer would be that due to solvation in $\ce{H2O}$, the $\ce{Cl-}$ ions would get stabilised and hence their basic strength would reduce as compared to those ions which are dissolved in $\ce{DMSO}$.


Am I right in saying this or did I overlook something?


