What is your travel date ? Otherwise we won't be able to help.

There are convenient direct TGVs between Lille and Strasbourg avoiding the Paris transfer. Limited 10€ fare, then 20€.

From Strasbourg 200 km/h TER trains run every 30 min to Basel (seat reservation not possible).

For journeys within France, use this website (4€ booking fee per order compared to 2€ per person per train on eurail.com) : https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish

Aiming for Friday14th July. There are seats I can book full price on SNCB  international but there is no place to put my pass in. The French website won’t recognize Basel as a station (I think because it’s not in France). 
 

and the eurail website is so unstable it cannot complete any bookings 

Please use the complete URL as given above, NOT the regular B-Europe website.

I will try on the computer tomorrow because it doesn’t work on my phone. Still doesn’t recognize Basel as a station when I use that link provided above 

Please read @thibcabe's post. You only need to book Lille to Strasbourg. From there, you just take a reservation-free fast regional train.

It only works for journeys within France as explained above. I’d choose this good connection via Strasbourg :

• TGV Lille-Europe - Strasbourg 15:52 - 18:51 20€ 2nd or 1st class (incl. upgrade)
• TER Strasbourg - Basel SBB 19:21 - 20:38

If you need to arrive earlier, you’ll have to go via Paris (more expensive). For example :

• TGV Lille-Flandres - Paris-Nord 11:12 - 12:14 10€
• transfer (not included)
• TGV Paris Gare de Lyon - Basel SBB 14:22 - 17:26 29€

The last TGV is available on eurail.com or by calling SNCF.

Thank you. The eurail website is so unstable I haven’t met anyone who has successfully booked on it 

I will make that booking in the morning