The majority of the burden for record keeping falls on the campaign committee, under the auspices of the campaign treasurer, but the ultimate responsibility is with the candidate and the donor.
You, as a citizen, are restricted in the amount you can provide. By the FEC's own rulebook, the committee must use its best efforts to source the gift:
If you contribute more than $200 to a committee, the committee is
required to use its best efforts to collect and publicly disclose on a
financial report your name, address, occupation and employer, as well
as the date and amount of your contribution. Committees sometimes
request this information even for smaller contributions, since the
$200 reporting threshold applies to your total contributions to one
committee during a calendar year. For example, you may make several
small contributions to a committee during a year. Once these
contributions add up to over $200, the committee must report the
Often, it is simply safer to not accept (or return) money that could not be properly sourced. Imagine you were a foreign national, federal contractor, or a minor: all prohibited from donating. The obscured donation comes in, the committee cashes the checks and moves on, and then nine months later a story comes about about fraudulent campaign donations in violation of FEC law.
The small loss in donations is generally nothing compared to the liability of accepting donations from potentially illegal channels. Further, allowing for donations and not bothering to ask identifying information certainly does not meet the bar of "best efforts to collect" information.