Via BEDOPS tools and UCSC data, you can:

1) Get SNPs for your reference genome:

$ wget -qO- http://hgdownload.cse.ucsc.edu/goldenPath/hg19/database/snp142Common.txt.gz \
    | gunzip -c - \
    | cut -f2,3,4,5,10 - \
    | awk -v OFS="\t" '{ print $1, $2, ($2 + 1), $4, $5 }' - \
    | sort-bed - \
    > hg19.snp142.bed

2) Get gene annotations; for example, from Gencode:

$ wget -qO- ftp://ftp.sanger.ac.uk/pub/gencode/Gencode_human/release_19/gencode.v19.annotation.gff3.gz \
    | gunzip -c - \
    | gff2bed - \
    | awk '($8="gene" && $4!~/^exon/)' - \
    | cut -f1-6 - \
    > hg19.gencode19.genes.bed

3) Do a bedmap operation to map hg19 SNPs within 1kb of hg19 gene annotations:

$ bedmap --echo --echo-map-id-uniq --delim '\t' --range 1000 hg19.gencode19.genes.bed hg19.snp142.bed > answer.bed

The file answer.bed contains genes and all rs* IDs of SNPs that fall within 1000 upstream or downstream of each gene interval.