I checked some Arabic ornaments and frame images with Google. They had generally sharp corners and smooth curves. This obviously is an enlargened version of a pixelated shape which originally has also contained a few corners and smooth curves.
Unfortunately no automate can decide which corners should be smoothed and which should stay, the only possibility for proper recreation without having the original is drawing. The draughtsman should have either some ingenuity or style knowledge.
I suggest you search some clipart and try to find a stylistically acceptable frame, if you cannot draw it. If you know what you exactly want to draw (=know well the right style), this should be done in a hour if you are a beginner or 10 minutes, if you have played a while with vector drawing software.
BTW. Here is one shape separated from your image (=black), a couple of different tracing attempts (=blue) and an attempt to draw it without actual knowledge of the wanted result (=red).
I'm afraid they all are useless. My drawing has a shortcut through all stairs plus some random deviation here and there due the inaccurate hand and eye. One should see in half a second it hasn't any actual visio of style.
Photoshop is as wise about the style, but it can make the shortcuts much better (=systematically in the same way). Try it, if you can accept the result. Do this:
Fill the frame with some color to keep it different from black and white:
select the fill color area and smooth the selection with Refine Edge:
Fill with the smoothed selection in a new layer:
You can copy and paste this to Inkscape or Illustrator for tracing and restoring the thickness by adding a stroke. I tried the stroke in Photoshop as a Layer Style:
The result is as crap as the preceding ones, I would say. Proper result is not possible otherwise than by having a style idea and then drawing that idea, not trying to force the parts to fit your staircase version. Here's an attempt to draw it as single stroke and without having any reference under the drawing at the same time: