Since I installed new rims my spare will not fit new set up.  Is it a safety issue if the spare is not in carrier in case of a rear end collision?

Not really. The spare isn't meant to be any part of the collision protection.

Just wondering why you can't keep it in the carrier? Just because you have new rims/tires doesn't mean the spare will no longer fit the car. Unless you went to some outrageous rim/tire size, the road wheels, should you have a flat, will fit the carrier....it is adjustable. 

One of the first things I did on all my vettes was remove the spare.

Second was lower the rear.

The new rims are the same as the old 17by 9.5 but the lug nuts are smaller.  thanks I will pull a wheel off and see if the spare fits the lugs.  they installed a kit that looks like it may have a shim?  thank you for your help. New rims are a direct fit for 1988 corvette with 17/9.5 lug nuts are smaller or appear to be and they are grooved a special socket was included to remove, they are not anti theft just chrome lug nuts that are grooved?

If the lug nuts are different, just carry 5 nuts that will fit the spare rim. Put them in a bag and store in one of the cubbys in the back. I do this on my '74 because the road wheels and spare take different lug nuts. 

Thanks I will do that. The spare is an original corvette donut tire and rim.  The size is T155/70 R17 Temporary 

Goodyear can you buy a new tire I checked once with town fair and they do not have any? Again thanks for the help!!!