I've heard both that the death penalty is applied in a racially discriminatory fashion, and that it's not. I decided to look up the facts on my own.
According to statistics from the Department of Justice, from 1976 to 2005, whites committed 45.8% of all homicides, and blacks committed 52.2%. (This number for whites included Hispanics, so the actual percentage for non-Hispanic whites was actually lower.) Yet from 1976 to 2008, of the 1099 people who were put to death in the United States, 57% were white, while 34% were black (and 7% were Hispanic). From this broad overview, it would seem the penalty discriminates against whites. (More on this later.)
Death penalty opponents sometimes quote the disparity in its application for interracial killings. From 1976 to 1995, 101 blacks were put to death for killing whites, whereas only 5 whites were put to death for killing blacks, a 20 to 1 ratio. For all interracial crimes of violence, there are roughly 10 times as many black on white crimes as the other way around. This would seem to indicate that twice as many blacks receive the death penalty for interracial homicides. Yet during the same period, there were 1.4 million cases of "robbery with injury" involving black criminals and white victims, and 68,000 cases of white criminals and black victims, a 21:1 ratio. This is the category of violent crime most closely associated with the type of aggravated murder which would most likely result in the death penalty, and is consistent with the first statistic. (Capital murder cases usually involve aggravating circumstances, which basically means that the murder is committed in conjunction with another felony.)
But why are more whites put to death overall when they commit fewer homicides? My gut tells me that whites are probably much more likely to commit premeditated murder, while blacks are more likely to commit the more impulsive homicides. (This is borne out by the statistic showing that 80% of the homicides by poison were committed by whites, whereas 56% of the homicides by handgun were committed by blacks; poison takes forethought and planning, handguns are more often involved in impulsive acts.) Also, whites are more often serial killers, a crime much more likely to result in the death penalty -- and deservedly so.
The disparities in age and sex were far more striking. Of the 1099 people put to death in those years, 1088, or 99%, were men, while only 11 were women. The current ratio of inmates on death row is 98.2% male vs. 1.8% female. Obviously, males are more violent than females, but by how much? I couldn't find comprehensive statistics for male vs. female killers, but I did find statistics for various individual years, and the range went from a low ratio of 5 to 1 male to female murders in 1976 to a high of 11 to 1 in 1995. The 1999 the ratio was more typical at 7 to 1. (I was surprised the ratios weren't higher.) In any case, in none of the years did the ratio approach the 99 to 1 ratio at which the killers were put to death. (Yet you never hear anyone criticize the death penalty for being "sexist.")
Another fact that stands out is the average age at conviction -- 27. Given that criminals under the age of 18 are almost always tried as juveniles, that makes for a very youthful group of killers, or at least a very youthful group given the death penalty. Of course, the most murderous cohort has always been males between the ages of 18 and 24; this is true of both blacks and whites. Given that this is the age at which testosterone runs highest, and given testosterone's correlation with violence, perhaps this should not be so surprising. (Interestingly, 5 of the 11 women executed were convicted after the age of 40; with them, testosterone obviously plays less of a role.)
As far as the gender difference between the way the penalty is applied, far more women kill out of a sense of being threatened, rightly or wrongly, than do men. (If a homicide is committed in self defense, it's not considered murder.) This may account for some of the disparity, but I suspect that most juries are also simply less willing to put a woman in the electric chair.
There are arguments against the death penalty other than "racism" which are harder to dispute, mostly because they are philosophical in nature. One such argument is that human life is sacred, and shouldn't be taken under any circumstances. That one can't be countered with logic.
Another argument is that capital punishment is cruel and unusual and therefore unconstitutional. The electric chair is certainly cruel (though usually less so than the method of killing the murderer himself used), though given its frequent use in the past thirty years, not all that unusual.
Some say that it is not consistently applied, and the capricious nature of its application should render it null. Currently it is allowed in thirty-seven states, and but not in the other thirteen or the District of Columbia. Should murderers in Massachusetts be treated with more leniency than those in Texas? Hard to justify that one; but then again, that's an argument for consistency, not its existence.
Some say that it makes us look barbaric because most Western countries don't allow it.
Certainly the number of people on death row who've been exonerated because of DNA evidence has to give anyone pause. (This, to me, is the best argument, and has turned me from being pro-death penalty into an agnostic on the issue; I may turn against it in the near future.) My guess is that the lack of DNA evidence does not necessarily indicate innocence, only the absence of definitive proof, and that some of the people who were so "exonerated" were in fact guilty. But cops do sometimes lie, and I'm sure that some of the people exonerated must also have been innocent.
In any case, some of these are good arguments. There's also no question that in the first half of the twentieth century the death penalty was applied in a very racist manner. But the argument that it's currently racist is simply not true.