Read through your writing aloud. Written and spoken word are very similar, although written word tends to be more formal through practice. Run-on sentences tend to be sentences that go on and on. Take a second look at any sentence that requires more than one breath to say. Even if it isn't a run-on sentence grammatically, you can probably break it down into smaller ideas.
Check that suspicious looking sentences are made up of only one independent clause and dependent clauses. An independent clause is a part of a sentence -- use commas to separate the parts of a sentence -- that contains a subject and a predicate that modifies the subject. For example, in the sentence "The dog is brown," "the dog" is the subject and it is modified by "is brown" to be a brown dog.
Use commas and FANBOYS to separate independent clauses in a single sentence. FANBOYS is an acronym that stands for "For-And-Nor-But-Or-Yet-So." The FANBOYS are conjunctions -- words that join clauses together -- that can be used to make your writing sound more natural. Avoid having more than one independent clause in a single sentence as a general practice. However, sometimes two independent clauses are closely related enough that they bear using together. For example, "Eric has a new car, but the engine is already cracked."