The initial question confuses the problem. In normal operation, there's no negative OP input, feedback pulls the non-inverting input to ground +/- a few millivolts which is well within specified LM358 operation. Problem is possible negative overload and latch-up, as discussed by dick_freebird.
In case of LM358, behavior with negative input current is detailed discussed in the datasheet. The device has even a maximum current rating for Vin < -0.3V of 50 mA, which is quite a lot. Most modern OPs have a clamp diode rating of 5 or 10 mA. In other words, there shouldn't occur destructive latch up if the voltage across the 1K resistor doesn't exceed 50V.
It's further discussed that the output voltage can behave irregularly during clamping action, under circumstances gain reversal may occur. If you want to avoid it, place an additional schottky clamp.
I'm referring to original National Semiconductor data sheet, don't know if there are cheap replicas that don't manage to avoid latch up.