Your description seems to be confusing two different things. Can you show an example of what you're talking about?
(It may not be a good idea to use an ordinary kernel density estimate if your random variable is discrete.)
You can get negative $x$-values ending up with some positive density from a kernel density estimate,
simply because of the way KDEs work (and there are ways to do something about that if that's a problem - e.g. see the discussion here).
However, negative density (density<0) would be when the density is below the x-axis, not to the left of the y-axis. That doesn't happen with a correctly implemented kernel density estimate, as long as the kernel itself is non-negative.
The reason why you get negative values getting positive density follows straight from the way a kernel density estimate works.
A KDE replaces each observation $x_i$ by a little hill of probability density, of area 1/n (the kernel, centered at $x_i$), and the density estimate at a point $x_0$ is the sum of all the little density-hills (each of which is centered at one of the data points), evaluated at $x_0$.
See the description and diagrams here for further explanation of what goes on.
Now when any data point is close to 0, some of the little hill around it will be below zero. That is, some of the density will be at negative values of $x$.
If you have an exact 0, half the contribution to the density estimate will be at negative $x$'s.
Can you suggest a link which shows the values close to zero will end up to negative density and why is this so?
Sure, if you mean "positive density at negative $x$", try here.
However, the density plot in R starts from minus 100.
What's the bandwidth?
Can you show us, rather than tell us? Can you give some small sample of data with the same issues you're describing?