I was intrigued to see how you make a basket out of pine cones, and so I looked at Jenni's reference, but it is about making a hanging basket for holding pine cones. I don't say it's impossible to make a basket out of pine cones, but it seems to me an odd thing to do. But pine cones are decorative, and it's quite plausible to display them in a hanging basket. I am pretty sure that is what this must refer to.

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Note added at 7 hrs (2012-10-21 16:53:28 GMT)
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The plot thickens! There are, in fact, four different things that "pinecone hanging basket" could mean, and in principle it could mean any of them:

1. A basket made of pine cones
2. A basket to hold pine cones
3. A basket shaped like a pine cone
4. A basket decorated with one or more pine cones or pine cone motifs.

All of these definitely exist and pictures of all of them can be found on the Internet. Modern pictures of baskets and videos of people making them are of no use at all in deciding what sort of object the narrator was expecting to find in this reception room in, presumably, the early twentieth century (since this is from a story by O. Henry).

1. Baskets made of pine cones did indeed exist in this era, and it is quite true that there is a craft tradition of objects made from natural materials. Here is an example of the right era:
"American, 19th and 20th century. Folksy pine mirror with pockets, candle sockets, and hanging rack. Wear. 24"h. together with a basket made of pine cones. 20"h."
Click on the image to enlarge it
http://www.artfact.com/auction-lot/mirror-and-basket.-1-c-17...

2. Pine cones as decorative objects have quite a long history and it was certainly customary in the nineteenth century to adorn the home with them, particularly at Christmas. This is certainly not a modern innovation.
"Holly was also used in the nineteenth century household for holiday decoration. All greenery would be adorned with berries, ribbon, wax, dough or metal ornaments and dried flowers. Pine cones were also scattered throughout the house."
http://jessicajewettonline.com/blog/page/90

"Follow Mary Beekman’s Diary each week to learn what it’s like to be a young child in early 19th century America [...] 'I hope the boys bring back some pine cones for the decorations for Christmas.'"
http://beekman1802.com/mary-and-cold-weather/?print=1

Given this fondness for decorating the home with pine cones, it is certainly quite possible that this hanging basket was a basket containing pine cones.

3. Baskets shaped like pine cones can also be found. Here's an example:
"Antique pine cone basket"
See the accompanying image
http://bostonconsigns.com/catalog/index.php?route=product/pr...

This seems to me a perfectly plausible possibility, and there's no reason to assume that it is less likely than the others.

4. A pine cone basket can certainly also mean a basket decorated with a pine cone motif. There are a number of references to antique ceramic pine cone baskets of this kind. Although it is perhaps less likely that they would have been used as hanging baskets, it is not impossible. Here's an example; there are a number of others:
http://pinterest.com/pin/95771929546283563/

So really it's impossible to be sure which sort of basket this refers to. What is certain, from many sources, is that the use of pine cones and images of pine cones for decorative purposes was very common at this time and had been since at least the eighteenth century.

As for Andrew Jackson, I really don't think there's any connection between him and the pine cone basket; the context simply implies that this is the kind of place where you would have expected to find a picture of Jackson and a basket of this kind. Jackson, the seventh President of the United States (1829-37), was associated with the Democratic Party, which was founded by followers of his.