How to counteract the deterioration of speech in stutterers by targeting sympathetic arousal?
The first line treatment for stuttering (I'm a stutterer myself) is speech therapy. Such therapies generally focus on helping stutterers to learn ways to minimize stuttering when they speak, such as by speaking more slowly, regulating their breathing, or gradually progressing from single-syllable responses to longer words and more complex sentences. Most of these therapies also help address the anxiety a person who stutters may feel in certain speaking situations (source: NIH). Anxiety management has especially worked well for me.

No drug therapy is currently approved against stuttering. I also don't know of any drug that targets the sympathetic nervous system. Having said that, there are a few interesting drugs investigated in clinical trials so far:


Dopamine-blocking medications against bipolar disorder, Tourette’s Disorder and Schizophrenia have been found to be effective in reducing the symptoms of stuttering (source: NSA). Their side effects, however, don't seem to weigh up against their effectiveness.
Pagoclone, a GABA-agonist used as anxiolytic has shown some effectiveness, but not enough to warrant it to be marked as anti-stutter therapy (source: BSA)