In population theory, there is a problem in explaining why sex continues to exist, and to "prove" their point, all creationists would have to do is cite the sentences from below - or sentences like them :
"The question of the adaptive value of sexual reproduction has proven to be the source of a great deal of controversy in evolutionary biology, and has occupied the minds of some of the big names in the field for over twenty years... The search for the adaptive value of sexual reproduction became yet more problematic when Maynard Smith (1975) proved that there is a two-fold reduction in fitness associated with sexual reproduction. This 'cost of males' occurs in aniosgamous species (species with gametes of different sizes). "
(The maintenance of sex, Graeme Pound, http://www.hpcc.ecs.soton.ac.uk/~gep98r/third.html)
However, it is hardly the problem that creationists make it out to be.
The central problem has to do with the cost of sex which is entirely incurred by the female, at least in most species. She is at a disadvantage compared to the males, so all things being equal, she would seem to be at a disadvantage when compared with the asexually reproducing female who passes on all of her genes to the next generation. From the perspective of population theory, it would seem that asexually-reproducing females would overrun the population - in the long-run.
Arguments which simply point to the survival value of sexual reproduction for the population are just that. They do not show how it is advantageous to the individual female in the short run - at the level that natural selection works - and as it is analyzed in terms of population theory. Nevertheless, many solutions have been proposed - with the real difficulty being that of finding empirical support for them. (The "problem" is given an additional level of complexity in that some species are capable of both sexual and asexual reproduction.)
At this point, evidence seems to be accumulating for the view that asexually-reproducing females are at a disadvantage in that their offspring are unable to occupy as many niches and are in fact limited to only one niche. Of course, there are other so-called "problems," such as explaining sexual dimorphism, etc, but typically when scientific literature cites sex as something problematic, this is what they mean.
It is not as if we have no solutions for explaining how sex is maintained given the paradox uncovered by population theory: we have about twenty of them. The problem is choosing between them on the basis of the evidence, and currently the evidence suggests that clonal populations will be much more limited in their niches - not to mention far more susceptible to diseases which have the potential for wiping out such monocultures. The genetic diversity made possible by sexual reproducing greatly mitigates such threats and opens up many more ecological opportunites due to the combinatorial power of sexual recombination.
I know it comments a joke but a serious answer is to maintain bonding - in species where offspring development is slow (humans e.g. par excellence) and need parental input then benefit for parents to stay together.
In social animals things that maintain group harmony are useful, hence sex as bond reinforcement tool.
Short term "cost" analysis is too simplistic as the whole point is long term survival which is aided by reasonable ability to change with conditions so genetic variety is useful.
Asexual reproduction CAN give genetic variety - mutations by radiation, chemicals etc, transcription errors etc all contribute however this "background" mutation rate is slow, but in organisms with short reproductive lifecycle i.e. lots of generations in a short time, then you can get some genetic changes. As has been mentioned though even "primitive" organisms have alternate strategies such as plasmid swaps by bacteria.
In longer reproductive lifecycle species asexual reproduction slow "background" rate of change is insufficient to give good genetic variety and so sexual reproduction is more likely to give long term competitive advantage.
Note there are also considerations of number of offspring; high numbers can give some defence against lower "fitness" in crowding out competitors. Thus there are cases when both sexual and asexual reproduction may be useful e.g. some plants, aphids etc. as ability to get large numbers of "clones" quickly can allow exploitation of a resource that has just become available and can help crowd out competitors and these organisms (usually) also have the sexual route open so they have the option of producing "fitter" offspring too.
But, as a basic rule, longer reproductive lifecycle organisms normally tend to do better with sexual reproduction as it gives more diversity and better chance of coping with change