The short answer is: a great deal of influence, directly and indirectly. We have already touched on this in section B.2.3 ("How does the ruling class maintain control of the state?") Here we will expand on those remarks.
State policy in a capitalist democracy is usually well-insulated from popular influence but very open to elite influence and money interests. Let's consider the possibility of direct influence first. It's obvious that elections cost money and that only the rich and corporations can realistically afford to take part in a major way. Even union donations to political parties cannot effectively compete with those from the business classes. For example, in the 1972 US presidential elections, of the $500 million spent, only about $13 million came from trade unions. The vast majority of the rest undoubtedly came from Big Business and wealthy individuals. For the 1956 elections, the last year for which direct union-business comparisons are possible, the contributions of 742 businessmen matched those of unions representing 17 million workers. And this was at a time when unions had large memberships and before the decline of organised labour.
Therefore, logically, politics will be dominated by the rich and powerful -- in fact if not in theory -- since only the rich can afford to run and only parties supported by the wealthy will gain enough funds and favourable press coverage to have a chance (see section D.3, "How does wealth influence the mass media?"). Even in countries with strong union movements which support labour-based parties, the political agenda is dominated by the media. As the media are owned by and dependent upon advertising from business, it is hardly surprising that independent labour-based political agendas are difficult to follow or be taken seriously. Moreover, the funds available for labour parties are always less than those of capitalist supported parties, meaning that the ability of the former to compete in "fair" elections is hindered. And this is ignoring the fact that the state structure is designed to ensure that real power lies not in the hands of elected representatives but rather in the hands of the state bureaucracy (see section J.2.2) which ensures that any pro-labour political agenda will be watered down and made harmless to the interests of the ruling class.
To this it must be added that wealth has a massive indirect influence over politics (and so over society and the law). We have noted above that wealth controls the media and their content. However, beyond this there is what can be called "Investor Confidence," which is another important source of influence. If a government starts to pass laws or act in ways that conflict with the desires of business, capital may become reluctant to invest (and may even disinvest and move elsewhere). The economic downturn that results will cause political instability, giving the government no choice but to regard the interests of business as privileged. "What is good for business" really is good for the country, because if business suffers, so will everyone else.
David Noble provides a good summary of the effects of such indirect pressures when he writes firms "have the ability to transfer production from one country to another, to close a plant in one and reopen it elsewhere, to direct and redirect investment wherever the 'climate' is most favourable [to business]. . . . [I]t has enabled the corporation to play one workforce off against another in the pursuit of the cheapest and most compliant labour (which gives the misleading appearance of greater efficiency). . . [I]t has compelled regions and nations to compete with one another to try and attract investment by offering tax incentives, labour discipline, relaxed environmental and other regulations and publicly subsidised infrastructure. . . Thus has emerged the great paradox of our age, according to which those nations that prosper most (attract corporate investment) by most readily lowering their standard of living (wages, benefits, quality of life, political freedom). The net result of this system of extortion is a universal lowering of conditions and expectations in the name of competitiveness and prosperity." [Progress Without People, pp. 91-92]
And, we must note, even when a country does lower its standard of living to attract investment or encourage its own business class to invest (as the USA and UK did by means of recession to discipline the workforce by high unemployment), it is no guarantee that capital will stay. US workers have seen their companies' profits rise while their wages have stagnated and (in reward) hundreds of thousands have been "down-sized" or seen their jobs moved to Mexico or South East Asia sweatshops. In the far east, Japanese, Hong Kong, and South Korean workers have also seen their manufacturing jobs move to low wage (and more repressive/authoritarian) countries such as China and Indonesia.
As well as the mobility of capital, there is also the threat posed by public debt. As Doug Henwood notes, "[p]ublic debt is a powerful way of assuring that the state remains safely in capital's hands. The higher a government's debt, the more it must please its bankers. Should bankers grow displeased, they will refuse to roll over old debts or to extend new financing on any but the most punishing terms (if at all). The explosion of [US] federal debt in the 1980s vastly increased the power of creditors to demand austere fiscal and monetary policies to dampen the US economy as it recovered . . . from the 1989-92 slowdown." [Wall Street, pp. 23-24] And, we must note, Wall street made a fortune on the debt, directly and indirectly.
Commenting on Clinton's plans for the devolution of welfare programmes from Federal to State government in America, Noam Chomsky makes the important point that "under conditions of relative equality, this could be a move towards democracy. Under existing circumstances, devolution is intended as a further blow to the eroding democratic processes. Major corporations, investment firms, and the like, can constrain or directly control the acts of national governments and can set one national workforce against another. But the game is much easier when the only competing player that might remotely be influenced by the 'great beast' is a state government, and even middle-sized enterprise can join in. The shadow cast by business [over society and politics] can thus be darker, and private power can move on to greater victories in the name of freedom." [Noam Chomsky, "Rollback III", Z Magazine, March, 1995]
Economic blackmail is a very useful weapon in deterring freedom.