The Middle East is a vast region just like any other and safety could vary hugely between countries, cities, and even neighborhoods within a certain city. When considering working or interning, you definitely want to be selective about where you go. When I studied in university, many Arabic students would study or intern in Syria, but right now that's unfortunately no longer an option.
In terms of safety, it's best to consult both the US State Department's most up-to-date country information and locals from a country you're interested in visiting. Your university is a good place to ask around to find students, professors, or other contacts who either come from or know someone from the prospective country.
For example, when I received my internship offer to work in Cairo, I checked the US State Department for the latest travel warnings for Egypt and Cairo itself and then spoke with a few friends of friends who were Egyptian. I always take the State Department's listings with a grain of salt because you have to remember they need to mention every "what if" scenario to cover their backs, so their listings for most countries sound pretty alarming. It's important we understand that kidnappings, rapes, and murder happen in every country on the planet. What you should be looking for is information on general political stability, group in-fighting that may affect outsiders, targeted violence against foreigners/Americans, and widespread violent crime. Very few places fit that kind of description.
It's important to speak with locals (if you can) before you go because they can provide a nuanced perspective on what their country or city may look like on the news compared to what it's actually like living there. They can also hook you up with their friends or relatives so when you arrive you have some local contacts. For example, I had a friend living in Lebanon when a number of attacks were going on in Beirut. In his section of the city, people were going out to bars and walking to work and living normal day-to-day life. Just a few miles away, people were staying inside to avoid gunfire or even fleeing that part of the city. Local information was critical to understanding where in Beirut was safe to visit.
Lastly, where you intern in the Middle East should depend on your actual career interests. Tel Aviv is a widely celebrated entrepreneurial hub, so if you're interested in working at a start-up, you should be looking at Israel. If you're interested in diplomacy, Jordan is a great country because of close US-Jordanian ties. If you're studying education, Qatar is the regional hub of international education thanks to the work of the Qatar Foundation. The UAE would be good for tourism, oil, or banking, and Egypt is a large, important country with huge institutions in every sector.