How to know if blackberries are safe to eat?

As soon as you get them home, inspect for damaged berries & discard any that are not perfect. Wash the rest in a colander or sieve under cold running water. Drain well.

Spread on dry paper towel for 30 mins to fully drain & dry, then put them in a new box with paper towel under & over - don't seal them with a snap lid.

Place in the fridge, in the salad crisper if you have one.

Life-span should now be more like a week.

The rapid spread of mould is from possibly just one single damaged berry. If they are touching & sweating, that can spread like wildfire. If they are clean, dry & cold, in air that can circulate around them, then you should be fine.
You could possibly be OK eating ones from the opposite side of the original punnet, but I wouldn't really. Safe rather than sorry, so do the preventative work first.

If it happens again even after you took proper care, pick a new store, or complain at the first one - though bear in mind that even with care, they are still fragile & won't last forever. Your sign of end-of-life when clean & dry should be shrinkage, they will eventually start to dry out & look wrinkled, rather than mould.

From comments under the OP ^[1] - it's possible that the mould spores are in the fridge not in the berries. The berries, if damaged in any way, simply provide a suitable breeding ground.
I'd recommend a fridge-sanitisation day. My method will prevent fast cross-contamination, but won't kill existing spores.

^{[1] "It's very strange since this problem started only in the last 2 months. I don't know if the grocery store is continuously selling moldy berries or not"}

How am I supposed to store blackberries to prevent it from molding after 1 day?

Buy frozen.

1. They're fresher (picked ripe and flash frozen, versus picked unripe so they last to the grocery store).
2. You can defrost only the ones you need.

I wouldn't even try to squeeze a week out of blackberries. Basically they should be eaten the instant you get a hold of them, they're too fragile and not meant for storage. Tetsujin's answer would prolong their life by a few days, but not indefinitely. And any blackberries that lasted much longer would be so impregnated with preservatives that you should probably steer clear of them.

That said it's not at all unusual for a basket of soft berries, whether from the store or a market, to contain one or more that didn't even make it to you. I throw out any that are really moldy, eat the rest without ever having any sort of problem.

Don't use Lysol in your fridge, that's worse than any spores. Just clean it out well using baking soda as a scouring powder, making sure to clean any crannies, that for instance you pull out the drawers to clean under them. Or if you really have a raging mold problem, rinse it down with a borax solution, if you can get your hands on some (7-mule team brand in the US is usually sold with laundry).