It probably means it's fully booked. Night trains can be fully booked weeks in advance, especially in summer and around holidays. What's the travel date?

In any case, it's best to look for alternatives. You can also travel by day.

Also a possibility is to travel in the seat carriage that runs on the same train like nightjet this route, but they sell it as Intercity. Reservations are optional, but highly reccomened. You can reserve via bahn.com seat only option for 4,50€ or via ÖBB for 3€ with the seat only option.

I travel on May 1st. When I clicked on Bahn.com’s reservation impossible explanation it says “this may occur if the connection is inside a transit agency, short distances, or for trains with reservations required”. 
Is there another sleeper train I could take perhaps? 

So the other options would be :

• travel on the night train in a seat (uncomfortable for sleeping, lights fully on)
• spend a night in Cologne or somewhere else along the way
• 30th April : Nightjet Amsterdam Centraal - Basel SBB 94€ in a shared 2-people sleeper
• 2nd May : Nightjet Amsterdam Centraal - Basel SBB 34€ shared 6-people couchette compartment, 44€ shared 4-people couchette compartment, 74€ shared 3-people sleeper
• travel on 1st May (daytime). One of the many options :
• ICE Amsterdam Centraal - Basel Bad Bf (doesn’t run to Basel SBB that day) 07:38 - 14:36
• ICE Basel Bad Bf - Zürich HB 14:48 - 16:00

The last daytime train would leave Amsterdam at 15:24 (in Zurich between midnight and 1am). Multiple connections along the way

How do I book the night jet from Amsterdam to Basel on may 2nd?

nightjet.com or tickets.oebb.at

Add Eurail as a discount and it should give you those prices. Let us know if it doesn’t work

I’m typing “eurrail” under select discounts but nothing is popping up.

Go to tickets.oebb.at. There, add the Interrail/Eurail discount for all persons, and then select "One-way tickets and day tickets”. Do NOT use "Seat reservation only”.

Do you know if they offer sleepers for women only, not just couchettes?

Sleepers are gender-separated, except when you book a complete compartment.

Just booked! Thank you so much for the quick and useful replies!