Law and connectionsWatch
More law resources on TSR
For bigger firms, no. They don't. It's helpful to get your face recognised by HR and get good feedback during a vacation scheme, but that's about it.
For smaller firms, who knows, they'll want to get to know you, but that doesn't mean connections/nepotism come into play. 'Getting to know you' could just mean you paralegalling there first for 6 months.
Connections can be made at all levels through cold emailing, using Linkedin to get in touch with people, asking people to go for a coffee with you, etc.
These days, candidates who have family members who are practising lawyers benefit indirectly (e.g. through access to informal work experience and advice on what interviewers actually look for) rather than directly ("I got hired because daddy is a lawyer here"). The overwhelming majority of people I know who are practising did not have connections at the firm they are working at.
Don't know much about the Bar. Crazy Jamie is the expert!
Networking is a bit different. It is a useful thing to do, but not because if you hit it off with a barrister you'll be offered pupillage. Again, it may be a route into a mini pupillage or similar, but the use of networking with barristers is that it can help to give you that insight into the gap between being a student and a practising barrister, and what you need to do to bridge it (which I've said before is one of, if not the most important thing for a pupillage application). One of the best way to gain that information and insight into what you need to be a practising barrister is to be around practising barristers, and networking helps you to do that.
All that said, and to directly answer the question, no you don't need connections to become a barrister. As with anything having connections can help, but they're unlikely to be the one difference between getting pupillage and not and you can, in any event, forge those connections yourself if you're minded to.