How can you block backlink checking sites, like Majestic, Ahrefs and SEMRush, from crawling or indexing your website?

 Here you go:

Most important bots to block if you like to do:

Majestic SEO -> User-agent: MJ12bot
MOZ OpenSiteExplorer -> User-agent: rogerbot
Ahrefs -> User-agent: Ahrefs

Robots.txt:

User-agent: Rogerbot  
User-agent: Exabot  
User-agent: MJ12bot  
User-agent: Dotbot  
User-agent: Gigabot  
User-agent: AhrefsBot  
User-agent: BlackWidow 
User-agent: ChinaClaw  
User-agent: Custo  
User-agent: DISCo  
User-agent: Download\ Demon  
User-agent: eCatch  
User-agent: EirGrabber  
User-agent: EmailSiphon  
User-agent: EmailWolf  
User-agent: Express\ WebPictures  
User-agent: ExtractorPro  
User-agent: EyeNetIE  
User-agent: FlashGet  
User-agent: GetRight  
User-agent: GetWeb!  
User-agent: Go!Zilla  
User-agent: Go-Ahead-Got-It  
User-agent: GrabNet  
User-agent: Grafula  
User-agent: HMView  
User-agent: HTTrack  
User-agent: Image\ Stripper  
User-agent: Image\ Sucker  
User-agent: Indy\ Library 
User-agent: InterGET  
User-agent: Internet\ Ninja  
User-agent: JetCar  
User-agent: JOC\ Web\ Spider  
User-agent: larbin  
User-agent: LeechFTP  
User-agent: Mass\ Downloader  
User-agent: MIDown\ tool  
User-agent: Mister\ PiX  
User-agent: Navroad  
User-agent: NearSite  
User-agent: NetAnts  
User-agent: NetSpider  
User-agent: Net\ Vampire  
User-agent: NetZIP  
User-agent: Octopus  
User-agent: Offline\ Explorer  
User-agent: Offline\ Navigator  
User-agent: PageGrabber  
User-agent: Papa\ Foto  
User-agent: pavuk  
User-agent: pcBrowser  
User-agent: RealDownload  
User-agent: ReGet  
User-agent: SiteSnagger  
User-agent: SmartDownload  
User-agent: SuperBot  
User-agent: SuperHTTP  
User-agent: Surfbot  
User-agent: tAkeOut  
User-agent: Teleport\ Pro  
User-agent: VoidEYE  
User-agent: Web\ Image\ Collector  
User-agent: Web\ Sucker  
User-agent: WebAuto  
User-agent: WebCopier  
User-agent: WebFetch  
User-agent: WebGo\ IS  
User-agent: WebLeacher  
User-agent: WebReaper  
User-agent: WebSauger  
User-agent: Website\ eXtractor  
User-agent: Website\ Quester  
User-agent: WebStripper  
User-agent: WebWhacker  
User-agent: WebZIP  
User-agent: Wget  
User-agent: Widow  
User-agent: WWWOFFLE  
User-agent: Xaldon\ WebSpider  
User-agent: Zeus 
Disallow: / 

.htaccess:

SetEnvIfNoCase User-Agent .*rogerbot.* bad_bot 
SetEnvIfNoCase User-Agent .*exabot.* bad_bot 
SetEnvIfNoCase User-Agent .*mj12bot.* bad_bot 
SetEnvIfNoCase User-Agent .*dotbot.* bad_bot 
SetEnvIfNoCase User-Agent .*gigabot.* bad_bot 
SetEnvIfNoCase User-Agent .*ahrefsbot.* bad_bot 
SetEnvIfNoCase User-Agent .*sitebot.* bad_bot 
<Limit GET POST HEAD> 
Order Allow,Deny 
Allow from all 
Deny from env=bad_bot 
</Limit>