I am putting up a secondhand 2 metre 13 element Yagi I bought at a rally years ago. The dipole is brand new and never used, the other elements are undamaged and I have just replaced all the plated steel hardware with stainless steel fasteners. But how do I decide if it should have a balun, and if it should, what type please? ...
I'd agree that most beams need some sort of balun for good
performance. The question of what type, however, depends
on the design of the driven element and the antenna itself.
There are three basic types of driven elements: those where
the center is open to connect the feedline (like a common HF
dipole), those that use a continuous element with no gap, often
fed using a gamma match (sometimes called "plumbers delight"
construction because it can be made from copper pipe with
no insulators), and those that use a folded dipole or other
other multi-element design. (There are others as well - these
are the main ones.)
You also need to know the design impedance for the antenna
feedpoint: yagis can be designed anywhere in the range from
about 5 ohms to 75 ohms. Some can be used without any
impedance matching (though a balun is still a good idea) and
some are designed for 12.5 ohms to use with a 4 : 1 impedance
step-up in the balun (or from a folded dipole element).
DK7ZB likes to design his for 28 ohms, as that permits matching
with a quarter wave section made by paralleling two pieces of
75 ohm coax. Many designs come out in the 20 - 25 ohm range,
which can be matched to 50 ohms with a shunt coil (often a "hairpin").
Some commercial designs use the hairpin to transform the antenna
impedance up to 200 ohms, then use a 4 : 1 balun or folded dipole
to step it back down to 50 ohms.
So you really need to know how the original antenna was designed
to be fed. That will give us a better idea of what sort of balun to
recommend.
To the other question of "how do I know if I need a balun", one
answer is to put up the antenna, tune in a constant signal, and plot
the radiation pattern as you turn the antenna. Radiation from the
coax (which is what a balun is designed to prevent) will often
distort the pattern, though you might not notice it if the signal source
and antenna are both horizontal and the coax runs vertically.
(Installing the antenna for vertical polarization requires attention to
other details, such as keeping the coax and metal mast/tower out
of the radiation pattern.)