This is the age old question on the Camino.
If you are USED to wearing boots and have well-broken in boots, or if you have weak ankles, wear boots.
For me, boots are overkill.
The Camino is a long walk, not a mountain climbing expedition.
Trail runners have worked just fine for me all these years.
It's just a matter of preference.
Whatever shoe you wear, be sure it has enough of a deep, wide toebox that you can wiggle your toes freely.
And you'll want padding around that ankle.
Also take into consideration the fact that most people's feet will swell, meaning you need to buy boots/shoes larger than you generally wear.
This is GENERALLY (not always) so people don't throw apples at me.
Personally, I have seen more discarded boots along the Camino than any other item, except maybe sleeping pads.
I've also escorted many pilgrims to shoe shops to turn in their boots for trail runners.
So I vote for trail runners.