What is safer Military police or infantry?

For their wartime mission, them MPs manage rear area security. Mostly using armored HMMWVs with SAW, M2 or Mk19 weapons mounted. That puts them in the light/motorized infantry camp. (although what passes for 'rear-area' these days is iffy) For their peacetime mission, they do law enforcement on military installations with no 'infantry' component to it. You drive a government squad car and do traffic stops. Every now and again you get a domestic violence in the housing area to make the day exciting. FWIW, most police departments like 11-series retirees for their police force. You will not find an infantry MOS to be a handicap when looking for your civilian job. If you want infantry training and missions, you really need to consider an 11-series MOS.
MPs have been described as Infantry on wheels. MPs rarely do garrison duty (ie. traditional police functions on an Army post) but now do mostly combat support operations like guarding detainees, escorting convoys and patroling highways "over there". The US Army military police are not infantry. Their mission is not to close with and kill, capture, or destroy the enemy. It is possible that they might be called on to perform an infantry mission if the combat situation was so distratrous as to have them fight to keep from being over run.

So if you don't want to be in the foxhole, or in the cold don't join the Infantry. The Infantry is expert in close order combat to engage and defeat the enemy. MP's are the ones who patrol the street's, and keep order after an area has been secure.

Addendum:There was a saying that was common in the US Army when I served back in the dark ages: When all the missiles have been launched, when all rounds have been fired down range, when the Jeep is out of fuel, and the bridge has been laid, and chow has been served, every soldier in the United States Army is infantry.