The person who rents is behind the proverbial eight ball. You need all the same protection that the homeowner needs; yet you are not free to make the needed repairs without approval from the owner. As it may take several hundred dollars to get the place into a condition where you will feel safe, you need to check with the owner to see if he/she will provide the money.
You may need to become inventive in your effort to provide a safe home for your family. See if the owner will provide the necessary material, if you provide the work involved or if you pay to have the work done. If all else fails, ask the owner if it would be okay for you to buy the material and pay to have the work done.

As a last resort, buy lots of plastic sheeting and follow the steps for sealing the home. The only difference will be that you are doing it on a temporary basis on the inside. If the owner still has a problem with what you are doing for the safety of your family, then I suggest that you find someplace else to move to where you can live safely.