It's an interesting question. From the other side of the argument, some research commissioned by the ABI (which favours scrapping stamp duty) suggested that one effect of the current regime was to push investors into contracts-for-difference, thus giving them an exposure without the tax. The report argued that this was a negative for corporate governance, and you can see the logic of the argument even if you don't accept it.
But we are now in Big Crisis Land where speculation is EVIL, and it would be relatively straightforward to portray Tory policy as being a sop to the investment industry and the 'spivs and speculators'. I would also question whether it would actually benefit savers, since it might simply result in more trading, adding frictional costs which offset the abolished tax. So it strikes me that perhaps Labour ought to have a serious attack on Tory policy here.
Whether actually upping stamp duty would affect behaviour as we would like I'm not so sure. It may not lead fund managers to trade less, as they might simply pass on the new cost to clients unless it was set at a level which really stings. And at such a level would that actually help/hurt the right people? An alternative idea that has been floated a few times is providing an increased dividend for investors who hold a company's shares for a given period of time. This ought to provide a small incentive to think more long term. Fund managers have been hostile to this idea when I've run it by them in the past, but again we're in new territory now. However again is the incentive strong enough to change behaviour?
Duncan's other point - that actually fund managers' clients put pressure (intentionally or otherwise) on them for short-term performance - is key here. Unless various actors in the system collectively acknowledge that in aggregate the increased turnover of equities is both a) a cost rather than a benefit and b) bad news for governance not much may result from financial incentives or penalties in respect of trading. So, what are the trustees of your pension fund up to on this score?
As an afterthought this paper came out a few years ago and really blows a hole in the idea that either companies or shareholders really think of the relationship as an ownership one. Here's a chunk:
both managers and investors seem to accept that shareholders in general are in some sense owners of the firm, with concomitant rights and responsibilities. Managers accept that they have a general duty to manage their companies for the benefit of their shareholders, and that shareholders as a body can legitimately expect them to engage in a constructive discussion as to what that duty might entail. Investors for their part recognize that the shareholders of a company have a duty to watch over it and to ensure that it is competently managed. For both sides, however, these rights and duties seem to reflect general, almost theoretical features of the capitalist system as operated in America and Britain, and to refer to shareholders in the abstract. They are not apparently conceived as duties owed by or to any particular shareholders, either individually or collectively. The owner-fiduciary model, in other words, acts as an ideal description of the system but not as a real description of the situation with which the actors are in practice faced. In conceptualizing this real world of day-to-day practice, neither managers nor investors cast the latter as owners.