There's a lot of potential for good material here. On a small scale the Foundation would absolutely try to contain and suppress evidence of extra-terrestrials at the very least until such time as they could figure them out. For example, it's often been kicked around headcanon-wise that the Foundation contained coelacanths until they realized that there's nothing anomalous about them — they really did just survive from ancient eras on their own. Now that is common, uncontained knowledge.
But such a big world changing event as alien motherships appearing over every city? Well, that can't be contained. That kind of contact is also a pretty bad sign for the future of humanity in general, so the Foundation would probably react defensively to such a thing… as in, going deep underground, saving what they can to preserve humanity and watching from the shadows for some sign as to what these aliens want. I could even see a story that concluded by the Foundation triggering some kind of cataclysm to wipe out the aliens and life on Earth at the same time and then come out of hiding to repopulate or reset reality.
Anyway, fertile ground for ideas there, so go wild.