Parents are responsible for this type of situation. For courts to become involved in "protecting children from advertisements" would involve making rulings that would lead to laws and regulations that would be so completely unconstitutional and against our expectations for rights of expression and competition as to be impossible.

In the United States, regardless of early resistance, it is the norm that protection of classes and groups of people and species and environments and rights is the province of the Federal and State governments. In this country, Federal protections usually come, but they come only after (1) much resistance; (2) much outcry over the need for protection; (3) much evidence, of one sort or another, that protection is requisite (necessary). These sorts of protections limiting advertisers and shielding children will eventually come, and they will come from the Federal government. These protection acts will join a long line of protection acts that includes among other protections environmental protection, industrial waste protection, civil rights protection, and privacy protection.

It seems to be generally accepted that it's parent's responsibility to protect kids from advertising to a certain degree. When extreme elements like cigarettes or liquor are involved, it is more accepted that legal constraints control what can be shown or printed.

The media outlets themselves do some self-policing in these matters.

I think the bulk of the responsibility should on the parents.

As a parent, I would say that it is my job to protect my children from advertising that counter to my values.  With that being said, I do believe the government should play a role in regulating advertising.  I don't want to be driving down the road and have my children objected to an obsense billboard.  There are some things even the most diligent parent can't protect against.

Parents should be the first and primary protectors and teachers of their children. Though the government does have the responsibility to keep the country safe, it is not the government's right or responsibility to assume the role of parent for anyone. While it is perfectly appropriate for the government to place restrictions on some things for the greater public good, parents still have the privilege and right to be responsible for their children.

Unfortunately, we are living in a time when too many parents have abdicated their roles as protectors and providers, and the government has assumed that role in way too many ways.

Advertising works or companies would not spend money on it; therefore, parents need to teach their children to be discerning and make good choices. Your question assumes that all or most advertising is detrimental, and that is just not true. If children learn to reason, they will make their choices in life without being swayed too much or too far by any negative influences, including advertisements. 

I agree that ultimately parents are responsible for supervising and educating their children about the things they see in advertisements. It does seem reasonable, though, to place some restrictions on ads, in particular the time slot in which certain ads are allowed to run on television, or not allowing liquor companies, for example, to build billboards across from elementary schools. There is a difference between a "nanny state" and sensible restrictions on businesses that sell products that are inappropriate for children.

It's got to be the responsibility of the parents.  There is no way that the government should be in the business of "protecting" kids from advertising in general.  It has to be up to the parents as to how much advertising their kids should see (at home, at least) and what kinds of ads their kids will be allowed to watch.  It's pointless to have the courts try to do it since parents can circumvent that by letting their kids watch adult shows where the courts would surely not be censoring ads.

This is really a matter of opinion. First, I would say that in our current media-saturated society, it is simply not possible to prevent children from being exposed to advertising (or violence or pornography). It's all too easily accessible via too many different devices. Given that, I think it is the responsibility of parents and schools (and churches, for the religious) to educate children so that they will not be overwhelmed by peer pressure and advertising into becoming superficial and materialistic people.

The courts and other government organizations do have the responsibility for monitoring and preventing false and misleading advertising, though.

Thus it's really more of a both/and than either/or issue.