This simple example will help you:
# example data
df = data.frame(id = c("A","B"),
                date_ref = c("2013-01-26", "2013-01-08"),
                date1 = c("2013-01-23", "2013-01-01"),
                date2 = c("2013-01-20", "2013-01-07"),
                stringsAsFactors = F)
df
#   id   date_ref      date1      date2
# 1  A 2013-01-26 2013-01-23 2013-01-20
# 2  B 2013-01-08 2013-01-01 2013-01-07
library(dplyr)
library(lubridate)
library(tidyr)
# update date column to datetime variables
# (if needed)
df = df %>% mutate_at(vars(matches("date")), ymd)
df %>%
  gather(type,date_new,-id,-date_ref) %>%        # reshape dataset
  group_by(id) %>%                               # for each id
  filter(abs(difftime(date_ref, date_new)) == min(abs(difftime(date_ref, date_new)))) %>%  # keep row with minimum distance between dates
  ungroup() %>%                                  # forget the grouping
  select(-type) %>%                              # remove that variable
  inner_join(df, by=c("id","date_ref"))          # join back original dataset
# # A tibble: 2 x 5
#      id   date_ref   date_new      date1      date2
#   <chr>     <date>     <date>     <date>     <date>
# 1     A 2013-01-26 2013-01-23 2013-01-23 2013-01-20
# 2     B 2013-01-08 2013-01-07 2013-01-01 2013-01-07
Not sure if you can have multiple dates with the same distance (same date multiple times of same number of days before/after your baseline date) and how you want to treat them.