No, the level of your Pokémon does not directly increase the chance to successfully run away (as of gen 5+). However, one of the main factors is speed, as explained below, and obviously your stats, one of which being speed, increase as you level up, so level could indirectly increase your odds. This is why Pokémon of higher level are generally harder to run away from.
Source 1:
On this Bulbapedia article, it says that this is the equation to determine your chances to successfully escape:
Neither this nor the "other factors" section under it mentions the level of either Pokémon. See the linked "other factors" or the linked Pokébase posts in sources 2 and 3 for more information on what does affect the chances of running away successfully.
Souce 2/3:
Neither This post nor this post mentions the level effecting your chances of escaping successfully. The second one specifically mentions that it doesn't.
Source 4
Experience
Hope I helped!
