Not only is it known not to be the case, they're understanding more precisely just what the values were. Only scientists made of straw assume that the C14 levels have been constant.
Someone on TalkOrigins' feedback page made the same claim. This response was interesting:
Thanks for this information. This indeed interesting and significant work. After a bit of checking, I can give some further background.Your text is probably taken from a very recent article by Roger Highfield, a science editor for the Electronic Telegraph. It appears in the June 30, issue and is available on-line at the time of writing this response.
Dr Richards has a home page from which you can find some of his publications.
The work in question is published in the latest issue of Science, June 29, 2001, as "Extremely large variations of atmospheric 14C concentration during the last glacial period", by J.W. Beck et al.
One of Dr Richards' research interests is the calibration of radiobarbon dating. This involves checking the dates using other independent methods. The main source of systematic error in radiocarbon dating is the varying concentration of carbon-14 in the atmosphere in the past. Dr Richards and his colleagues have found good evidence of increased levels beyond about 30,000 years ago, with a very substantial peak around 43.3 to 44.3 thousand years ago, by studies of a stalagmite.
They consider increased cosmic radiation from a supernova as a possible explanation for increased generation of C14 in the atmosphere around that time.
There have been many other studies allowing calibration of atmospheric radiocarbon by various independent means; the study by Richards and his co-authors largely confirms and refines those calibrations; the finding of a large peak is a new result going beyond the limits of previous calibrations.
The large peak they found is a new and interesting result, though it has no effect except for dates extending back over 33 thousand years.
In summary, this work confirms the principles of radiocarbon dating, confirms and refines existing calibration of radiocarbon dates from 11,000 to something like 24,000 years, extends calibration back to 45,000 years, leading to some significant corrections for dates greater than 30,000 years. These corrections mean that some published dates may be too young. No major change for dates less than 30,000 years is indicated, and an interesting peak was found near the end of their range of study.
There is, of course, not the slightest comfort for young earth creationists in these results. There may be cause for some substantial adjustments to published dates for some studies, such as Neanderthal sites, making them a bit older than previously thought, by up to 20%.