markrb2021, Budapest is a very, very safe city. If someone told you the area is well patrolled, then I would question the source as visible patrols other than an ocassional police car driving thru on the way to someplace else, just isn't the norm in the city. Doesn't need to be. The Keleti area does border the less than fun part of District VIII (south of the station, maybe 4 blocks) and that part of District VIII is probably a place that would make many uncomfortable. But if you take the M2 from Keleti one stop into town you are clear of all of that.
Budapest isn't a restored jewell. Some areas maybe, but as a whole, no. But that's part of the attraction, it doesn't exist for the tourists to the same degree as the areas you would be inclined to visit in Vienna or Prague.
For your one morning convenience to the train, you will be spending nothing less than 45 minutes each day getting to and from tourism. Not horrible, and if you get creative with the routes to the town center, you will see a lot of amazing Budapest.
Around the station one building can be quite nice, the next ehhhhh. But again, not about safety, just its a poor country and the restoration tends to follow the money and the money isn't that far out yet.
Someone mentioned the Oktogon. If you stayed at the Oktogon you would take the 4 or 6 tram 3 stops to Blaha Lujza ter then the M2 to Keleti. 10 to 15 min tops. If you stayed at Deak Ferenc ter, 10 min on the M2 will get you to Keleti. The Keleti metro station is in the lower level of the train station. That easy.
Would I worry about staying near Keleti? No. Not at all. But I do want you to enjoy your stay. For some, where you are staying would be fine. Others might regret it because of the distance and because much is in "original" communist occupation condition. I can't make that decision for you.
If you have an address and you send it to me, maybe I can be a bit more precise because a block one way or the other can be different. I know a bit of the area because I go to some of the Asian markets and an Asian food court in the area as well as the station.