In that case, pretty much anything would work for the backing, but I agree with tesspug that duckcloth would be my first choice -- inexpensive, weighty & appropriately simple for an alter (and easy to launder). Honestly, my second choice would be either dyed or unbleached muslin -- I still wouldn't go with white unless your church has linens professionally cleaned or you don't expect the alter topper to be used for more than one season. I used to belong to a church that was so hard on its linens that the women were asked to make new linens for every liturgical season. It was insanely demanding and in the end people were taking shortcuts left and right because we knew they wouldn't last -- things like "self-binding" to a piece of cheap felt and always making the part that hung off the sides of the alter pointed & just folding it over on itself to form a triangle shape (since that's the one place the felt would have shown). The only stitching that was done other than decorative embroidery was just a single tight stitch to piece. No point in reinforcing since it was getting tossed in a couple months anyways.
If this is a piece that your church plans to use year after year, then perhaps someone could contribute a nice white linen tablecloth that you could cut apart & use as the backing. If it's going to be well cared for, it's worthwhile to put some time into sourcing materials & doing beautiful stitching. In that case, your idea of stitching 1/4" from the seam would be quite lovely (I'd still use a heavier thread or triple stitch it). With it being 2 layers of fabric, the more stitching you add through both layers, the nicer it will lay on the alter & the better it will hold up to laundering.