I've been using plastic cable connectors, as can be seen in the picture below, to hold a bicycle basket fixed to the rack.
I would look at 3~5 small aluminium fish plates in the basket, and small saddle clamps under the rails. You'd need two small stainless steel bolts and nylock nuts per plate, or some other way to reduce vibration.
Suggesting Stainless Steel to avoid rust, or you could just use grease. This will make undoing the bolts hard later, and rust may mark things you put in the basket, like cloth.
This suggestion would not be easier, but it will be a lot more durable.
There are metal zip ties, but they will have the same loosening problem, which is exacerbated by the small radius they have to clamp around.
You could use old spokes or other metal wire. Spokes are hard to bent with fingers so pliers might be required. Any sharp end could poke a hole into load items in the basket too. I've got a box of thousands of galvanised concrete reinforcing tiedown wires that can be bent by hand.
I have a fair bit of experience with these baskets, and can tell you with confidence that the mesh will not hold up to any force that exceeds that of a plastic zip-tie. However, you might just want to try heavier-duty plastic zip ties with a metal pawl. Make sure they are wrapped around the main frame of the basket, not just the mesh, and pull them as tight as you can. Also, if you insulate the contact points betwen the rack and the basket with a bit of rubber or leather (old inner tube works great), it will dampen a lot of the vibration that is probably causing the ties to come loose.
I would either use some dedicated set, like this one: Basil crate fixing set (picture comes from their webpage).
Or I would fabricate something similar using 4 penny washers (on top of the basket) with 4 M4 or M5 bolts with nuts and 2 metal (aluminium) strips for under the carrier rack.
Perhaps 2 bolts with penny washers and 1 metal strip along the rack would be enough.
While you can certainly improvise something using metal fasteners, you could also consider buying a dedicated basket fastener.
Ask your local bicycle store for a "basket fastener" or "basket holder" - they should be able to help you. The nice thing is that in addition to fasteners that are screwed in place, you can also get fasteners that can be detached by pressing a button or lever - that is useful to detach the basket if you want to use it for shopping, or if you temporarily need the rack for something else.
I personally use the KORBfix (non-detachable) by Rixen & Kaul (German company, but also sold e.g. in the US) . For more information, go to the manufacturer's site, and enter "korbfix" in the search box. Rixen & Kaul also offer a range of detachable fasteners, under the brand name KLICKfix.