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?
Well, a couple reasons:
Am also has a nice gamma line, so a better choice would have been plutonium. Your borated paraffin will not do anything against the gammas, a lead casket will do a little better.
2) Each smoke detector contains ~35 kBq of americium (according to Google), so you have approximately 4.7 E7 Bq, or a couple milliCi if you have 1350 smoke detectors (and if I can count). I do not know where you plan to play with it, but that's not stuff that you want to just handle on the kitchen table or a bench in your garage.
3) Beryllium? The element is incredibly toxic.
4) There are better choices for neutron moderators than borated paraffin.
5) A lead lined concrete vault is ok when the set-up is encased in it. But judging by the set-up that you have, it will not be safe outside of the vault.
I am sure that someone else can think of some more reasons, or correct any mistakes that I have made. Why are you going through all this trouble just to activate some silver? Sure, it is a cool experiment, but what will you do with everything when it is all over?