Your resume needs to make your strongest case. If the job you worked at for 6 months makes you look less qualified for the job you are seeking, then leave it off but be prepared to explain the gap in employment. It is rare that this would be the situation, though. Usually simply going to work every day, no matter where you go, looks better than a gap.
In the equally rare situation where your resume must be 1 page long, you may omit less qualifying jobs so that you can fit other qualifying factors such as capabilities, publications, or certifications onto the resume. Again, your resume must make your best case and nobody will blame you for making that happen, though they may follow up in an interview.
Finally, your title asked about background checks, even though the content of the question did not. Always provide exactly the information requested on a background check. If the check wants to know all of your jobs for the last 5 years, don't list one worked 6 years ago (it won't impress them) but you must list every single one in the last 5 years, even if you worked there for days. Background investigators assume all omissions are intentional and all omissions are lies.