Autocorrelated errors signal model misspecification. Ideally, model errors should be $i.i.d.$ and thus should have no patterns in them. If they do, there is some information left unextracted; some more modelling can be done to extract the pattern.

There are two ways of dealing with the problem of autocorrelated errors.

1. Leave the model specification as is but expand confidence intervals around the regression coefficients to account for the violation of the model assumption of non-autocorrelated errors. This can be motivated by the wish to retain the original model that may be directly derived from theory and/or have a nice interpretation. This can be done by using heteroskedasticity and autocorrelation (HAC) robust standard errors, e.g. by Newey and West (1987). HAC standard errors (as an alternative to the regular standard errors) should be available in any major statistical software package; they seem to be quite popular among practitioners, perhaps because they provide an easy solution.
   Pros: easy to use; can retain the original model.
   Cons: wider confidence intervals $\rightarrow$ lower precision, harder to reject null hypotheses; model is misspecified.
2. Change the model specification to obtain non-autocorrelated errors. For example, run a regression with ARMA errors (easy to implement by arima or auto.arima functions in R including the regressors via the parameter xreg) or -- as DJohnson suggested -- include lags of dependent variable as regressors.
   Pros: narrower confidence intervals $\rightarrow$ higher precision, easier to reject null hypotheses; model is correctly specified (unless there are other faults, which may quite often be true).
   Cons: requires more work; cannot retain the original model.

References:

• Newey, Whitney K; West, Kenneth D (1987). "A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix". Econometrica 55 (3): 703–708. doi:10.2307/1913610

Autocorrelated errors signal model misspecification. Ideally, model errors should be $i.i.d.$ and thus should have no patterns in them. If they do, there is some information left unextracted; some more modelling can be done to extract the pattern.

There are two ways of dealing with the problem of autocorrelated errors.

1. Leave the model specification as is but expand confidence intervals around the regression coefficients to account for the violation of the model assumption of non-autocorrelated errors. This can be motivated by the wish to retain the original model that may be directly derived from theory and/or have a nice interpretation. This can be done by using heteroskedasticity and autocorrelation (HAC) robust standard errors, e.g. by Newey and West (1987). HAC standard errors (as an alternative to the regular standard errors) should be available in any major statistical software package; they seem to be quite popular among practitioners, perhaps because they provide an easy solution.
   Pros: easy to use; can retain the original model.
   Cons: wider confidence intervals $\rightarrow$ lower precision, less power (harder to reject null hypotheses); model is misspecified; less accurate forecasting (due to neglecting the autocorrelation in model errors).
2. Change the model specification to obtain non-autocorrelated errors. For example, run a regression with ARMA errors (easy to implement by arima or auto.arima functions in R including the regressors via the parameter xreg) or -- as DJohnson suggested -- include lags of dependent variable as regressors.
   Pros: narrower confidence intervals $\rightarrow$ higher precision, more power (easier to reject null hypotheses); model is correctly specified (unless there are other faults, which may quite often be true); more accurate forecasting.
   Cons: requires more work; cannot retain the original model.

I side with Francis Diebold's forceful argumentation (in his blog post "The HAC Emperor has no Clothes") that 2. is the way to go.

References:

• Newey, Whitney K; West, Kenneth D (1987). "A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix". Econometrica 55 (3): 703–708. doi:10.2307/1913610