Can someone elaborate on this problem?
Yes, SPV mining allows to mine with just the previous block header. Transactions on the previous block aren't really checked.
A transaction that isn't balanced, uses inputs that have already been spent, or is otherwise malformed and doesn't meet the rules.
You include one of those in your block and your block will not be accepted by clients, and any blocks after it will not be accepted by clients.
Maybe accepted by light clients, but not by full clients. Run a full validating client to prevent such blocks from being accepted by your client and you are safe.
Don't run a full validating client and yes, this is one of many GIGO attacks you are vulnerable to.