It sould be safe, I'll pour a lead-lined concrete vault. Oh, and I did do the calculations; it would take approx. 1351.351351 Sd (smoke detectors, aka Microcuries) to get 1000 neutrons a second. And why are my materials not the best choices for this setup?
If I take Grejaks number of 35kBq per smoke detector, then your 1350 detectors would add up to about 1.3 milli-Curie of activity. The customary limit for safe handling of actinide elements in a well ventilated fume hood is 10 mico-Curie.
That means you are already a factor of 130 above this limit, and I don’t assume you have a good fume hood in your basement. How about a glove box?
These limits are there for a good reason. If a radioactive element emits an alpha particle, the remaining nucleus is pushed in the opposite direction. This can mean that the radioactive element literally starts to wander around and can contaminate the surroundings. I have worked with high level activities of alpha emitters and I have seen it happen. If you start messing with this much Am-241 there is a very good chance that you will contaminate the room you are working in and inhale some of it yourself.
I hate to say this, but this is a classic case of: Don't try this at home!