H. Davis wrote:
"How will it improve reception relative to just the whip (which is kind
of useless in the basement anyway)?

Your coax can be used as a transmission line, instead as part of the
antenna, to bring above ground signals into your basement while
shielding against noise inside your house if connected to a balanced
antenna somewhat removed from noise sources. Noise will appear as
common-mode in many cases balancing out in a coupling coil.

For a receiving antenna, a small inexpensive toroidal ferrite core
appropriate to the frequency of your receiver may be used on which to
wind a primary of an antenna coupling transforner. The secondary will be
the whip antenna itself.

The 19th edition of the ARRL Antenna Book shows a noise bridge for 1.8
to 30 MHz on page 27-24. The bridge generator is connected to the bridge
detector (receiver) and to the unknown impedance (antenna) via a
broadband toroidal transformer. This core would be right to couple coax
to a whip antenna except for its size and shape. It is a small binocular
core. All that`s needed to couple to a receiver whip antenna is a
ferrite ring with enough diameter for 3 or 4 turns of # 30 or so
enameled wire leaving enough hole space to be impaled over either of the
two whips.

The core used in the noise bridge is an Amidon BLN-43-2402. The material
is obviously apropriate for 1.8 to 30 MHz. I don`t have an Amidon
catalog handy but for sure the material is available in a simple ring.

Simply wind the coil around the ring and connect it your coax. Connect a
balanced antenna to the other end of the coax. The antenna should be
placed away from noise sources and placed in the field of desired
signals. Slip the ring over the whip of the receiver you want to use
with an external antenna. That`s it! You may need a balun at the
balanced antenna for minimum noise.

The whip forms a complete circuit through its capacitance to the earth
and to the radio circuit board or chassis. It is a 1-turn secondary on
the toroidal antenna transformer. Its operation is akin to an a-c
clamp-on ammeter.

This system has been used to couple a transmitter to a grounded
transmitting tower. For transmission, air was used for the core of a
large toroidal coil surrounding the tower. To tightly couple the coil to
the tower, the coil`s reactance was tuned out with a series variable
capacitor. This has a disadvantage of the tuned circuit type. It has a
narrow bandpass. For reception, you likely don`t need tight coupling as
gain is usually surplus in your receiver. You need signal to noise
ratio.

Toroidal coupling systems are not found in standard broadcasting
stations because the large toroid would be expensive, and difficult to
construct and insulate. It would also slice away the medium-wave
higher-frequency sidebands, limiting audio frequency response of the
station.

Untuned, the toroid coil coupling is tight enough for a high gain
receiver and bandwidth is sufficient for a 1.8 to 30 MHz noise bridge or
receiver.

Best regards, Richard Harrison, KB5WZI