Do Delta Basic Economy passengers get a carry-on?

On most Delta routes, yes. What I learned:

• You can generally bring one carry-on bag and one personal item.
• The personal item must fit under the seat in front of you.
• Basic Economy usually means no free checked bag on domestic flights, and limited seat selection.
• Rules can change on certain international routes, so check your itinerary.

Always measure your bag before leaving home. Here is a clear page on Delta carry-on rules.