It is not a run-on sentence.
It could be described as sentence fragment in some contexts, but sentences without the subject and verb contained in the typical sentence are often acceptable and desirable.
The example sentence may appear to be missing a subject and verb, but since that information can be taken from the context, good writers and educated or expert authorities would generally accept it, even in formal writing.
We can extract the subject and verb from the previous sentence:
[Such a person will self-strengthen] not for selfish or base pleasures, but to enrich both his or her own life and, more importantly, to enrich the lives of his loved ones, community, and the world.
Can we have effective and acceptable sentences without an explicit subject and verb? Certainly! Should we listen to people who say that a sentence must have a subject, or a verb, or both? No.
Wikipedia's current entry for sentence invites us to consider the first three sentences of Bleak House by Charles Dickens, which offer a challenge to commonly understood definitions of the word:
London. Michaelmas term lately over, and the Lord Chancellor sitting in Lincoln's Inn Hall. Implacable November weather.
There are, however, two apparent problems with the sentence. First, with the word both. It's likely not to be immediately clear, for many readers, what both refers to. Secondly, we have his or her in one phrase, but only his in another phrase, which doesn't seem to make sense.