As above, any brand focusing its efforts on eco-friendly fabrics such as organic cotton, hemp, vegan leather, natural rubber, and recycled materials is a good place to start. You also can't go wrong with a Certified B Corp - this means they've undergone rigorous testing to ensure they are actually sustainable.
Other brands are off-setting their footprint by taking back your old trainers to recycle them, and planting a tree for every pair sold. Ethical running shoe brand Allbirds are making their models carbon neutral by creating fabrics from Forest Stewardship Council-certified eucalyptus trees, and their trademarked SweetFoam used in midsoles is made with carbon-negative green EVA material.
While it doesn't seem as important, the packaging your trainers comes in is obviously key. Sending eco-friendly shoes in a plastic bag just isn't going to cut it, which is why some labels are also focusing on recycled cardboard packaging that is free of single-use plastic.