Yes cherry juice does have the ability to thin your blood.
If you're taking blood thinners then drinking too much cherry juice or eating too many cherries could cause your blood to get too thin which can cause other health issues.
Drinking some cherry juice or eating some cherries even when you are taking some blood thinners is okay as long as you do not eat too many cherries or drink too much cherry juice.
Cherry Juice as well as Cherries contain a natural plant compound known as quercetin which can also interact with some medications.
So if you're taking medications especially the blood thinners then speak with your doctor about it to make sure the cherries or cherry juice won't cause interactions with your medications.