Does going to a graduate school with less prestige than your undergraduate school hurt you?
Edited heavily based on further information provided by OP
If you are going into industry, or pursuing a Master's, it may hurt you in the short term
A Master's degree is mostly course work. It's too short for you to create a body of work that stands on its own. The quality of the courses largely determines, consequently, the quality of the Master's. Prospective employers may use an institution's reputation as a heuristic/proxy for the quality teaching.
Attending a less prestigious institution will make it more difficult to get interviews. Not being able to attend the bigger names also means you miss out on their alumni networks.
Create a compensatory professional network that overcomes these limitations. Most Master's in CS programs have a semester-long project. Make an impressive project and shop it around. This is a good way to build rapport with potential employers.