Richard made a good summary about the problem with common mode currents.
I'm a little leery, however, of the suggestion of putting a bead every
foot. The reason is that a common mode choke (aka current balun, or
bead) has low loss if its impedance is either much higher or much lower
than the common mode impedance. Loss is maximized when it's roughly the
same. So I can see situations where the distributed beads might end up
dissipating a significant fraction of the applied power. But I've never
set up and measured such a system, or modeled one, so don't have any
evidence this would happen.

As the developer and seller of antenna modeling software, I also caution
people that models of the types of baluns and transformers commonly used
for OCF dipoles are greatly in error when used as an OCF feed. The
problem is that on many bands the transformer or balun is operating with
impedances well outside its design range, and consequently adds series
and shunt impedance and exhibits a changed transformation ratio.

Here are a couple of postings I made on this forum some time ago:

3/10/07

Here are some characteristics of off-center fed dipoles which I've
observed in doing careful measurements:

1. The feepoint balun is commonly a voltage balun, which may have the
nominal claimed impedance transformation ratio over most of the HF range
when terminated with 50 + j0 ohms times the tranformation ratio.
However, the antenna doesn't have this impedance at hardly any
frequency, and can be very different at some frequencies. When presented
with load impedances typical of the antenna, the transformation ratio is
way off and becomes complex, and the balun adds considerable shunt and
series reactance.

2. Whenever a voltage balun is used to feed an asymmetrical antenna, it
creates an imbalance current in its attempt to equalize the voltages at
the two halves relative to the "cold" side of the input. This imbalance
current flows down the feedline as a common mode current.

3. Additional common mode current results from the unequal mutual
coupling between the feedline and unequal antenna halves.

4. It takes very concentrated efforts to reduce the common mode current
to a low level on all bands. Multiple current baluns (probably what the
CW calls and "isolator") are required, and even then it might also
require feedline length adjustments to get low common mode current on
all bands.

5. Without being able to quantify what the feedpoint balun will do in
terms of transformation, reactance, and common mode current generation,
it's impossible to build a model of one of these antennas with any
confidence, even if the feedline is included in the model. The best
efforts I made to measure a real antenna and its balun and build a model
based on the measurements led to generally poor agreement between the
measured and model impedance. Consequently I'm extremely skeptical of
any model that purports to predict anything about OCF dipole performance.

9/22/08

A while back I did some pretty careful measurements of an OCF dipole. I
found that ferrites were required at both the feedpoint and at one or
more places along the feedline. The ferrites at the feedpoint suppress
the conducted common mode current (which is actually forced to exist by
the voltage balun). But the asymmetry of the antenna results in common
mode current being induced onto the feedline by mutual coupling to the
antenna. This isn't a problem in a symmetrical dipole if the feedline is
positioned symmetrically relative to the antenna, since the currents
induced by the two equal halves cancel. But the OCF dipole can result in
quite a lot of induced common mode current.

Ideally, you'd put at least a second bunch of snap on cores about a
quarter wavelength from the feedpoint. But one of the main reasons
people use OCFs is for multi-band operation. So the thing to do is to
place the cores for maximum effectiveness on the band(s) where you have
the most trouble -- the common mode current also depends on the feedline
length and position, and will vary considerably from band to band even
if you do nothing.

My opinion is that users of OCF dipoles are just about always going to
have to deal with some amount of common mode current, and the best you
can do is reduce it to a level you can tolerate.

Roy Lewallen, W7EL