There are several known and conjectured mechanisms.
Best known is heterozygote advantage - where the mutation is deleterious in the homozygote condition, but advantageous in the heterozygous. The classical case is sickle-cell anaemia, where homozygous individuals have poor survival probabilities, whilst heterozygous individuals have resistance against malaria. (I think that Linus Pauling was involved in this discovery).
This is good enough perhaps if all you want is some explanation in principle that these diseases c
an survive. Less commonly told I think is that that example is one of the few clear-cut cases. From this example I think that when geneticists look at genetic disease distributions in populations they strongly suspect there
must be some heterozygous advantage. Supporting of this, for many of them there are different actual mutations affecting the same protein found in the affected individuals - e.g. If I remember there are a dozen or two different mutations affecting the gene for the enzyme phenylalanine hydroxylase, originating in different times and places, that cause the disease phenylketonuria in European populations. But then it is another matter to actually identify the advantage. So if you look up some of the common most common genetic diseases you find that there are suspicions, researches, but not a complete certainty for the moment. Possibly heterozygous defective gene that in homozygus individuals causes cystic fibrosis conferred resistance to infective disease for example
it is a fascinating area of research, or even just to read up. Many of the common disease mutations can be identified as having originated a matter of centuries or a few millennia ago. (E.g. about a millennium or less for common sickle-cell mutations.)
For their maintenance note that since none of them are exactly 'common', elimination by selection is slow. Rough calculations can often be made.
Just from memory fairly common diseases like sickle-cell anaemia would be predicted to halve in about a couple of centuries if the population is moved from where malaria is endemic, and this is held to correspond with its lower incidence in America than in the ancestral West African populations. After that, since the frequency of diseased individuals is the square of the damaged gene frequency, you can see the gene and dsease is eliminated ever more slowly, i.e. For nature to halve it again would take approx. twice as long as the first halving. So it is not too surprising that there is a relative handful of 'common' genetic disease, but thousands of 'rare' ones.
Another mechanism for maintenance is hitchhiking – suppose a deleterious mutation is close on the chromosome to another one which carries an advantage, so that both are usually inherited together, then the selection advantage of the one may compensate the disadvantage of the other.
Another mechanism is "mutation pressure". There are at least a few cases of this in the case of X-linked diseases, such as Lechs-Nyan syndrome. Selection pressure against is much more severe than in the case of heterozygous, since one half of the male offspring of a female carrier are affected. So if these diseases are unfortunately at significant levels, it is because new mutations are common. If I remember well about a quarter of the cases of that disease are caused by new mutations.
Already alluded to in particular examples, the history of populations is an essential part of the subject (and help to make it fascinating). You will find much in the literature of medical and population genetics about for example Québecois, Afrikaners, Ashkenazi Jews, Sardinians, Rom, and other populations with a special history, that all have diseases common in their population which are not so outside it.
Note that when we talk of "genetic disease" we are just talking about striking extreme though scientifically significant cases - those with a simple identifiable inheritance pattern. You could say that all disease is genetic in a sense. Beginning to be disentangled is "multifactorial disease" or disease predisposition - you may have, say, eight genes and several alleles thereof, certain combinations of which will be good and other combinations will Increase your likelihood of suffering from say heart disease. So things like "heterozygous advantage" are just special cases which Science could earlier identify because of their simplicity of something more general.
I am just writing this from memory and off the top of my head, as I am away from my books at the moment (and both my memory and my books are out of date!
)