Read the sentence aloud in your mind and make note of what it's about --- who the subject is and what information is being conveyed. Remember that around 20% of the time the sentence will be correct without any modification (option "A" of the five possible options).
Go through the four possible corrections one at a time. Say them aloud in your head --- if an option sounds awkward, discard it immediately and move to the next one. This will help you to narrow down your choices.
Look for errors in the possible corrections that remain. These may be subtle, so check to see if they make the subject of the sentence unclear, if they change the tense, if the meaning of the sentence appears to have changed, or if the correction introduces a change in "voice" --- whether a sentence is active or passive.
Double-check your final choice by rereading the entire sentence with your correction in mind. Make sure that it's clear and unambiguous, and that it doesn't seem unwieldy or strange --- if it sounds right, don't second-guess yourself.
Read the questions that follow the passage before you read the passage itself. This will help give you an idea of what to look for.
Skim the passage and create a mental outline --- note what is being talked about, what argument is being made and what information is being used to support that argument.
Reread the passage carefully, looking for information about the question you're answering this time. Remember that any information required in the question will be present in the passage --- so if an option seems to depend on information you're not given, throw it out.
Go through the possible responses one by one, looking for statements that can't clearly be supported by the passage, that depend on assumptions that are not spelled out in the passage or that appear to suggest a different argument than the one made by the passage. This process of elimination will remove obviously wrong answers.