This verse must be read in light of the context of the third chapter of Romans (and actually in light of the over-arching context of Romans itself). Paul is talking about righteousness through the law vs. righteousness through Christ. He begins by describing the life of one attempting to please God through legalistic righteousness (Romans 3:9-19). But Paul concludes that the end result is that we only condemn ourselves further, for we show our sinfulness to be truly sinful. Then, in 20ff, Paul describes the life lived by faith and the righteousness that comes through Christ. Immediately, he writes vs. 23, not to show that the life of the Christian is to be one of continual sin and failure, but rather to show that the righteousness which the believer enjoys is based upon Christ's grace, not upon the best legalistic efforts of the person trying to please God.
As anyone can plainly see, to use 3:23 as "proof" that Paul believed that Christians will necessarily continue to sin throughout their lifetimes is eisigesis--completely ignoring the context of Paul's meaning and message.
Now I am not saying, conversely, that this text is proof for the possibility of the Christian living a life without sin. Rather, I am only pointing out that it cannot be utilized either way.